Formation of a field reversed configuration for magnetic and electrostatic confinement of plasma

ABSTRACT

A system and method for containing plasma and forming a Field Reversed Configuration (FRC) magnetic topology are described in which plasma ions are contained magnetically in stable, non-adiabatic orbits in the FRC. Further, the electrons are contained electrostatically in a deep energy well, created by tuning an externally applied magnetic field. The simultaneous electrostatic confinement of electrons and magnetic confinement of ions avoids anomalous transport and facilitates classical containment of both electrons and ions. In this configuration, ions and electrons may have adequate density and temperature so that upon collisions they are fused together by nuclear force, thus releasing fusion energy. Moreover, the fusion fuel plasmas that can be used with the present confinement system and method are not limited to neutronic fuels only, but also advantageously include advanced fuels.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application is also related to provisional U.S. applicationSer. No. 60/266,074, filed Feb. 1, 2001, and provisional U.S.application Ser. No. 60/297,086, filed on Jun. 8, 2001, whichapplications are fully incorporated herein by reference.

[0002] This invention was made with Government support under ContractNo. N00014-99-1-0857, awarded by the Office of Naval Research. Somebackground research was supported by the U.S. Department of Energy for1992 to 1993. The Government has certain rights in this invention.

FIELD OF THE INVENTION

[0003] The invention relates generally to the field of plasma physics,and, in particular, to methods and apparati for confining plasma. Plasmaconfinement is particularly of interest for the purpose of enabling anuclear fusion reaction.

BACKGROUND OF THE INVENTION

[0004] Fusion is the process by which two light nuclei combine to form aheavier one. The fusion process releases a tremendous amount of energyin the form of fast moving particles. Because atomic nuclei arepositively charged—due to the protons contained therein—there is arepulsive electrostatic, or Coulomb, force between them. For two nucleito fuse, this repulsive barrier must be overcome, which occurs when twonuclei are brought close enough together where the short-range nuclearforces become strong enough to overcome the Coulomb force and fuse thenuclei. The energy necessary for the nuclei to overcome the Coulombbarrier is provided by their thermal energies, which must be very high.For example, the fusion rate can be appreciable if the temperature is atleast of the order of 10⁴ eV—corresponding roughly to 100 milliondegrees Kelvin. The rate of a fusion reaction is a function of thetemperature, and it is characterized by a quantity called reactivity.The reactivity of a D-T reaction, for example, has a broad peak between30 keV and 100 keV.

[0005] Typical fusion reactions include:

D+D→He³(0.8 MeV)+n(2.5 MeV),

D+T→α(3.6 MeV)+n(14.1 MeV),

D+He³→α(3.7 MeV)+p(14.7 MeV), and

p+B ¹¹→3α(8.7 MeV),

[0006] where D indicates deuterium, T indicates tritium, αindicates ahelium nucleus, n indicates a neutron, p indicates a proton, Heindicates helium, and B¹¹ indicates Boron-11. The numbers in parenthesesin each equation indicate the kinetic energy of the fusion products.

[0007] The first two reactions listed above—the D-D and D-Treactions—are neutronic, which means that most of the energy of theirfusion products is carried by fast neutrons. The disadvantages ofneutronic reactions are that (1) the flux of fast neutrons creates manyproblems, including structural damage of the reactor walls and highlevels of radioactivity for most construction materials; and (2) theenergy of fast neutrons is collected by converting their thermal energyto electric energy, which is very inefficient (less than 30%). Theadvantages of neutronic reactions are that (1) their reactivity peaks ata relatively low temperature; and (2) their losses due to radiation arerelatively low because the atomic numbers of deuterium and tritium are1.

[0008] The reactants in the other two equations—D-He³ and p-B¹¹—arecalled advanced fuels. Instead of producing fast neutrons, as in theneutronic reactions, their fusion products are charged particles. Oneadvantage of the advanced fuels is that they create much fewer neutronsand therefore suffer less from the disadvantages associated with them.In the case of D-He³, some fast neutrons are produced by secondaryreactions, but these neutrons account for only about 10 per cent of theenergy of the fusion products. The p-B¹¹ reaction is free of fastneutrons, although it does produce some slow neutrons that result fromsecondary reactions but create much fewer problems. Another advantage ofthe advanced fuels is that the energy of their fusion products can becollected with a high efficiency, up to 90 per cent. In a direct energyconversion process, their charged fusion products can be slowed down andtheir kinetic energy converted directly to electricity.

[0009] The advanced fuels have disadvantages, too. For example, theatomic numbers of the advanced fuels are higher (2 for He³ and 5 forB¹¹). Therefore, their radiation losses are greater than in theneutronic reactions. Also, it is much more difficult to cause theadvanced fuels to fuse. Their peak reactivities occur at much highertemperatures and do not reach as high as the reactivity for D-T. Causinga fusion reaction with the advanced fuels thus requires that they bebrought to a higher energy state where their reactivity is significant.Accordingly, the advanced fuels must be contained for a longer timeperiod wherein they can be brought to appropriate fusion conditions.

[0010] The containment time for a plasma is Δt=r²/D, where r is aminimum plasma dimension and D is a diffusion coefficient. The classicalvalue of the diffusion coefficient is D_(c)=a_(i) ²/τ_(ie), where a_(i)is the ion gyroradius and τ_(ie) is the ion-electron collision time.Diffusion according to the classical diffusion coefficient is calledclassical transport. The Bohm diffusion coefficient, attributed toshort-wavelength instabilities, is D_(B)=(1/16)a_(i) ²Ω_(i), where Ω_(i)is the ion gyrofrequency. Diffusion according to this relationship iscalled anomalous transport. For fusion conditions,D_(B)/D_(c)=(1/16)Ω_(i)τ_(ie)≈10⁸, anomalous transport results in a muchshorter containment time than does classical transport. This relationdetermines how large a plasma must be in a fusion reactor, by therequirement that the containment time for a given amount of plasma mustbe longer than the time for the plasma to have a nuclear fusionreaction. Therefore, classical transport condition is more desirable ina fusion reactor, allowing for smaller initial plasmas.

[0011] In early experiments with toroidal confinement of plasma, acontainment time of Δt≈r²/D_(B) was observed. Progress in the last 40years has increased the containment time to Δt≈1000r²/D_(B). Oneexisting fusion reactor concept is the Tokamak. The magnetic field of aTokamak 68 and a typical particle orbit 66 are illustrated in FIG. 5.For the past 30 years, fusion efforts have been focussed on the Tokamakreactor using a D-T fuel. These efforts have culminated in theInternational Thermonuclear Experimental Reactor (ITER), illustrated inFIG. 7. Recent experiments with Tokamaks suggest that classicaltransport, Δt≈r²/D_(c), is possible, in which case the minimum plasmadimension can be reduced from meters to centimeters. These experimentsinvolved the injection of energetic beams (50 to 100 keV), to heat theplasma to temperatures of 10 to 30 keV. See W. Heidbrink & G. J. Sadler,34 Nuclear Fusion 535 (1994). The energetic beam ions in theseexperiments were observed to slow down and diffuse classically while thethermal plasma continued to diffuse anomalously fast. The reason forthis is that the energetic beam ions have a large gyroradius and, assuch, are insensitive to fluctuations with wavelengths shorter than theion gyroradius (λ<a_(i)). The short-wavelength fluctuations tend toaverage over a cycle and thus cancel. Electrons, however, have a muchsmaller gyroradius, so they respond to the fluctuations and transportanomalously.

[0012] Because of anomalous transport, the minimum dimension of theplasma must be at least 2.8 meters. Due to this dimension, the ITER wascreated 30 meters high and 30 meters in diameter. This is the smallestD-T Tokamak-type reactor that is feasible. For advanced fuels, such asD-He³ and p-B¹¹, the Tokamak-type reactor would have to be much largerbecause the time for a fuel ion to have a nuclear reaction is muchlonger. A Tokamak reactor using D-T fuel has the additional problemthat-most of the energy of the fusion products energy is carried by 14MeV neutrons, which cause radiation damage and induce reactivity inalmost all construction materials due to the neutron flux. In addition,the conversion of their energy into electricity must be by a thermalprocess, which is not more than 30% efficient.

[0013] Another proposed reactor configuration is a colliding beamreactor. In a colliding beam reactor, a background plasma is bombardedby beams of ions. The beams comprise ions with an energy that is muchlarger than the thermal plasma. Producing useful fusion reactions inthis type of reactor has been infeasible because the background plasmaslows down the ion beams. Various proposals have been made to reducethis problem and maximize the number of nuclear reactions.

[0014] For example, U.S. Pat. No. 4,065,351 to Jassby et al. discloses amethod of producing counterstreaming colliding beams of deuterons andtritons in a toroidal confinement system. In U.S. Pat. No. 4,057,462 toJassby et al., electromagnetic energy is injected to counteract theeffects of bulk equilibrium plasma drag on one of the ion species. Thetoroidal confinement system is identified as a Tokamak. In U.S. Pat. No.4,894,199 to Rostoker, beams of deuterium and tritium are injected andtrapped with the same average velocity in a Tokamak, mirror, or fieldreversed configuration. There is a low density cool background plasmafor the sole purpose of trapping the beams. The beams react because theyhave a high temperature, and slowing down is mainly caused by electronsthat accompany the injected ions. The electrons are heated by the ionsin which case the slowing down is minimal.

[0015] In none of these devices, however, does an equilibrium electricfield play any part. Further, there is no attempt to reduce, or evenconsider, anomalous transport.

[0016] Other patents consider electrostatic confinement of ions and, insome cases, magnetic confinement of electrons. These include U.S. Pat.No. 3,258,402 to Farnsworth and U.S. Pat. No. 3,386,883 to Farnsworth,which disclose electrostatic confinement of ions and inertialconfinement of electrons; U.S. Pat. No. 3,530,036 to Hirsch et al. andU.S. Pat. No. 3,530,497 to Hirsch et al. are similar to Farnsworth; U.S.Pat. No. 4,233,537 to Limpaecher, which discloses electrostaticconfinement of ions and magnetic confinement of electrons with multipolecusp reflecting walls; and U.S. Pat. No. 4,826,646 to Bussard, which issimilar to Limpaecher and involves point cusps. None of these patentsconsider electrostatic confinement of electrons and magnetic confinementof ions. Although there have been many research projects onelectrostatic confinement of ions, none of them have succeeded inestablishing the required electrostatic fields when the ions have therequired density for a fusion reactor. Lastly, none of the patents citedabove discuss a field reversed configuration magnetic topology.

[0017] The field reversed configuration (FRC) was discoveredaccidentally around 1960 at the Naval Research Laboratory during thetapinch experiments. A typical FRC topology, wherein the internal magneticfield reverses direction, is illustrated in FIG. 8 and FIG. 10, andparticle orbits in a FRC are shown in FIG. 11 and FIG. 14. Regarding theFRC, many research programs have been supported in the United States andJapan. There is a comprehensive review paper on the theory andexperiments of FRC research from 1960-1988. See M. Tuszewski, 28 NuclearFusion 2033, (1988). A white paper on FRC development describes theresearch in 1996 and recommendations for future research. See L. C.Steinhauer et al., 30 Fusion Technology 116 (1996). To this date, in FRCexperiments the FRC has been formed with the theta pinch method. Aconsequence of this formation method is that the ions and electrons eachcarry half the current, which results in a negligible electrostaticfield in the plasma and no electrostatic confinement. The ions andelectrons in these FRCs were contained magnetically. In almost all FRCexperiments, anomalous transport has been assumed. See, e.g., Tuszewski,beginning of section 1.5.2, at page 2072.

SUMMARY OF THE INVENTION

[0018] To address the problems faced by previous plasma containmentsystems, a system and apparatus for containing plasma are hereindescribed in which plasma ions are contained magnetically in stable,large orbits and electrons are contained electrostatically in an energywell. A major innovation of the present invention over all previous workwith FRCs is the simultaneous electrostatic confinement of electrons andmagnetic confinement of ions, which tends to avoid anomalous transportand facilitate classical containment of both electrons and ions. In thisconfiguration, ions may have adequate density and temperature so thatupon collisions they are fused together by the nuclear force, thusreleasing fusion energy.

[0019] In a preferred embodiment, a plasma confinement system comprisesa chamber, a magnetic field generator for applying a magnetic field in adirection substantially along a principle axis, and an annular plasmalayer that comprises a circulating beam of ions. Ions of the annularplasma beam layer are substantially contained within the chambermagnetically in orbits and the electrons are substantially contained inan electrostatic energy well. In one aspect of one preferred embodimenta magnetic field generator comprises a current coil. Preferably, thesystem further comprises mirror coils near the ends of the chamber thatincrease the magnitude of the applied magnetic field at the ends of thechamber. The system may also comprise a beam injector for injecting aneutralized ion beam into the applied magnetic field, wherein the beamenters an orbit due to the force caused by the applied magnetic field.In another aspect of the preferred embodiments, the system forms amagnetic field having a topology of a field reversed configuration.

[0020] Also disclosed is a method of confining plasma comprising thesteps of magnetically confining the ions in orbits within a magneticfield and electrostatically confining the electrons in an energy well.An applied magnetic field may be tuned to produce and control theelectrostatic field. In one aspect of the method the field is tuned sothat the average electron velocity is approximately zero. In anotheraspect, the field is tuned so that the average electron velocity is inthe same direction as the average ion velocity. In another aspect of themethod, the method forms a field reversed configuration magnetic field,in which the plasma is confined.

[0021] In another aspect of the preferred embodiments, an annular plasmalayer is contained within a field reversed configuration magnetic field.The plasma layer comprises positively charged ions, whereinsubstantially all of the ions are non-adiabatic, and electrons containedwithin an electrostatic energy well. The plasma layer is caused torotate and form a magnetic self-field of sufficient magnitude to causefield reversal.

[0022] In other aspects of the preferred embodiments, the plasma maycomprise at least two different ion species, one or both of which maycomprise advanced fuels.

[0023] Having a non-adiabatic plasma of energetic, large-orbit ionstends to prevent the anomalous transport of ions. This can be done in aFRC, because the magnetic field vanishes (i.e., is zero) over a surfacewithin the plasma. Ions having a large orbit tend to be insensitive toshort-wavelength fluctuations that cause anomalous transport.

[0024] Magnetic confinement is ineffective for electrons because theyhave a small gyroradius—due to their small mass—and are thereforesensitive to short-wavelength fluctuations that cause anomaloustransport. Therefore, the electrons are effectively confined in a deeppotential well by an electrostatic field, which tends to prevent theanomalous transport of energy by electrons. The electrons that escapeconfinement must travel from the high density region near the nullsurface to the surface of the plasma. In so doing, most of their energyis spent in ascending the energy well. When electrons reach the plasmasurface and leave with fusion product ions, they have little energy leftto transport. The strong electrostatic field also tends to make all theion drift orbits rotate in the diamagnetic direction, so that they arecontained. The electrostatic field further provides a cooling mechanismfor electrons, which reduces their radiation losses.

[0025] The increased containment ability allows for the use of advancedfuels such as D-He³ and p-B¹¹, as well as neutronic reactants such asD-D and D-T. In the D-He³ reaction, fast neutrons are produced bysecondary reactions, but are an improvement over the D-T reaction. Thep-B¹¹ reaction, and the like, is preferable because it avoids theproblems of fast neutrons completely.

[0026] Another advantage of the advanced fuels is the direct energyconversion of energy from the fusion reaction because the fusionproducts are moving charged particles, which create an electricalcurrent. This is a significant improvement over Tokamaks, for example,where a thermal conversion process is used to convert the kinetic energyof fast neutrons into electricity. The efficiency of a thermalconversion process is lower than 30%, whereas the efficiency of directenergy conversion can be as high as 90%.

[0027] Other aspects and features of the present invention will becomeapparent from consideration of the following description taken inconjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Preferred embodiments are illustrated by way of example, and notby way of limitation, in the figures of the accompanying drawings, inwhich like reference numerals refer to like components.

[0029]FIGS. 1A and 1B show, respectively, the Lorentz force acting on apositive and a negative charge.

[0030]FIGS. 2A and 2B show Larmor orbits of charged particles in aconstant magnetic field.

[0031]FIG. 3 shows the {right arrow over (E)}×{right arrow over (B)}drift.

[0032]FIG. 4 shows the gradient drift.

[0033]FIG. 5 shows an adiabatic particle orbit in a Tokamak.

[0034]FIG. 6 shows a non-adiabatic particle orbit in a betatron.

[0035]FIG. 7 shows the International Thermonuclear Experimental Reactor(ITER).

[0036]FIG. 8 shows the magnetic field of a FRC.

[0037]FIGS. 9A and 9B show, respectively, the diamagnetic and thecounterdiamagnetic direction in a FRC.

[0038]FIG. 10 shows the colliding beam system.

[0039]FIG. 11 shows a betatron orbit.

[0040]FIGS. 12A and 12B show, respectively, the magnetic field and thedirection of the gradient drift in a FRC.

[0041]FIGS. 13A and 13B show, respectively, the electric field and thedirection of the {right arrow over (E)}×{right arrow over (B)} drift ina FRC.

[0042]FIGS. 14A, 14B and 14C show ion drift orbits.

[0043]FIGS. 15A and 15B show the Lorentz force at the ends of a FRC.

[0044]FIGS. 16A and 16B show the tuning of the electric field and theelectric potential in the colliding beam system.

[0045]FIG. 17 shows a Maxwell distribution.

[0046]FIGS. 18A and 18B show transitions from betatron orbits to driftorbits due to large-angle, ion-ion collisions.

[0047] FIGS. 19 show A, B, C and D betatron orbits when small-angle,electron-ion collisions are considered.

[0048]FIGS. 20A, 20B and 20C show the reversal of the magnetic field ina FRC.

[0049]FIGS. 21A, 21B, 21C and 21D show the effects due to tuning of theexternal magnetic field B₀ in a FRC.

[0050]FIGS. 22A, 22B, 22C and 22D show iteration results for a D-Tplasma.

[0051]FIGS. 23A, 23B, 23C, and 23D show iteration results for a D-He³plasma.

[0052]FIG. 24 shows iteration results for a p-B¹¹ plasma.

[0053]FIG. 25 shows an exemplary confinement chamber.

[0054]FIG. 26 shows a neutralized ion beam as it is electricallypolarized before entering a confining chamber.

[0055]FIG. 27 is a head-on view of a neutralized ion beam as it contactsplasma in a confining chamber.

[0056]FIG. 28 is a side view schematic of a confining chamber accordingto a preferred embodiment of a start-up procedure.

[0057]FIG. 29 is a side view schematic of a confining chamber accordingto another preferred embodiment of a start-up procedure.

[0058]FIG. 30 shows traces of B-dot probe indicating the formation of aFRC.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0059] An ideal fusion reactor solves the problem of anomalous transportfor both ions and electrons. The anomalous transport of ions is avoidedby magnetic confinement in a field reversed configuration (FRC) in sucha way that the majority of the ions have large, non-adiabatic orbits,making them insensitive to short-wavelength fluctuations that causeanomalous transport of adiabatic ions. For electrons, the anomaloustransport of energy is avoided by tuning the externally applied magneticfield to develop a strong electric field, which confines themelectrostatically in a deep potential well. Moreover, the fusion fuelplasmas that can be used with the present confinement process andapparatus are not limited to neutronic fuels only, but alsoadvantageously include advanced fuels. (For a discussion of advancedfuels, see R. Feldbacher & M. Heindler, Nuclear Instruments and Methodsin Physics Research, A271(1988)JJ-64 (North Holland Amsterdam).)

[0060] The solution to the problem of anomalous transport found hereinmakes use of a specific magnetic field configuration, which is the FRC.In particular, the existence of a region in a FRC where the magneticfield vanishes makes it possible to have a plasma comprising a majorityof non-adiabatic ions.

[0061] Background Theory

[0062] Before describing the system and apparatus in detail, it will behelpful to first review a few key concepts necessary to understand theconcepts contained herein.

Lorentz Force and Particle Orbits in a Magnetic Field

[0063] A particle with electric charge q moving with velocity {rightarrow over (v)} in a magnetic field {right arrow over (B)} experiences aforce {right arrow over (F)}_(L) given by $\begin{matrix}{{\overset{\rightarrow}{F}}_{L} = {q{\frac{\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}}{c}.}}} & (1)\end{matrix}$

[0064] The force {right arrow over (F)}_(L i)s called the Lorentz force.It, as well as all the formulas used in the present discussion, is givenin the gaussian system of units. The direction of the Lorentz forcedepends on the sign of the electric charge q. The force is perpendicularto both velocity and magnetic field. FIG. 1A shows the Lorentz force 30acting on a positive charge. The velocity of the particle is shown bythe vector 32. The magnetic field is 34. Similarly, FIG. 1B shows theLorentz force 30 acting on a negative charge.

[0065] As explained, the Lorentz force is perpendicular to the velocityof a particle; thus, a magnetic field is unable to exert force in thedirection of the particle's velocity. It follows from Newton's secondlaw, {right arrow over (F)}=m{right arrow over (a)}, that a magneticfield is unable to accelerate a particle in the direction of itsvelocity. A magnetic field can only bend the orbit of a particle, butthe magnitude of its velocity is not affected by a magnetic field.

[0066]FIG. 2A shows the orbit of a positively charged particle in aconstant magnetic field 34. The Lorentz force 30 in this case isconstant in magnitude, and the orbit 36 of the particle forms a circle.This circular orbit 36 is called a Larmor orbit. The radius of thecircular orbit 36 is called a gyroradius 38.

[0067] Usually, the velocity of a particle has a component that isparallel to the magnetic field and a component that is perpendicular tothe field. In such a case, the particle undergoes two simultaneousmotions: a rotation around the magnetic field line and a translationalong it. The combination of these two motions creates a helix thatfollows the magnetic field line 40. This is indicated in FIG. 2B.

[0068] A particle in its Larmor orbit revolves around a magnetic fieldline. The number of radians traveled per unit time is the particle'sgyrofrequency, which is denoted by Ω and given by $\begin{matrix}{{\Omega = \frac{q\quad B}{mc}},} & (2)\end{matrix}$

[0069] where m is the mass of the particle and c is the speed of light.The gyroradius a_(L) of a charged particle is given by $\begin{matrix}{{a_{L} = \frac{v_{\bot}}{\Omega}},} & (3)\end{matrix}$

[0070] where ν_(⊥) is the component of the velocity of the particleperpendicular to the magnetic field.

[0071] {right arrow over (E)}×{right arrow over (B)} Drift and GradientDrift

[0072] Electric fields affect the orbits of charged particles, as shownin FIG. 3. In FIG. 3, the magnetic field 44 points toward the reader.The orbit of a positively charged ion due to the magnetic field 44 alonewould be a circle 36; the same is true for an electron 42. In thepresence of an electric field 46, however, when the ion moves in thedirection of the electric field 46, its velocity increases. As can beappreciated, the ion is accelerated by the force q{right arrow over(E)}. It can further be seen that, according to Eq. 3, the ion'sgyroradius will increase as its velocity does.

[0073] As the ion is accelerated by the electric field 46, the magneticfield 44 bends the ion's orbit. At a certain point the ion reversesdirection and begins to move in a direction opposite to the electricfield 46. When this happens, the ion is decelerated, and its gyroradiustherefore decreases. The ion's gyroradius thus increases and decreasesin alternation, which gives rise to a sideways drift of the ion orbit 48in the direction 50 as shown in FIG. 3. This motion is called {rightarrow over (E)}×{right arrow over (A)} drift. Similarly, electron orbits52 drift in the same direction 50.

[0074] A similar drift can be caused by a gradient of the magnetic field44 as illustrated in FIG. 4. In FIG. 4, the magnetic field 44 pointstowards the reader. The gradient of the magnetic field is in thedirection 56. The increase of the magnetic field's strength is depictedby the denser amount of dots in the figure.

[0075] From Eqs. 2 and 3, it follows that the gyroradius is inverselyproportional to the strength of the magnetic field. When an ion moves inthe direction of increasing magnetic field its gyroradius will decrease,because the Lorentz force increases, and vice versa. The ion'sgyroradius thus decreases and increases in alternation, which gives riseto a sideways drift of the ion orbit 58 in the direction 60. This motionis called gradient drift. Electron orbits 62 drift in the oppositedirection 64.

[0076] Adiabatic and Non-Adiabatic Particles

[0077] Most plasma comprises adiabatic particles. An adiabatic particletightly follows the magnetic field lines and has a small gyroradius.FIG. 5 shows a particle orbit 66 of an adiabatic particle that followstightly a magnetic field line 68. The magnetic field lines 68 depictedare those of a Tokamak.

[0078] A non-adiabatic particle has a large gyroradius. It does notfollow the magnetic field lines and is usually energetic. There existother plasmas that comprise non-adiabatic particles. FIG. 6 illustratesa non-adiabatic plasma for the case of a betatron. The pole pieces 70generate a magnetic field 72. As FIG. 6 illustrates, the particle orbits74 do not follow the magnetic field lines 72.

[0079] Radiation in Plasmas

[0080] A moving charged particle radiates electromagnetic waves. Thepower radiated by the particle is proportional to the square of thecharge. The charge of an ion is Ze, where e is the electron charge and Zis the atomic number. Therefore, for each ion there will be Z freeelectrons that will radiate. The total power radiated by these Zelectrons is proportional to the cube of the atomic number (Z³).

[0081] Charged Particles in a FRC

[0082]FIG. 8 shows the magnetic field of a FRC. The system hascylindrical symmetry with respect to its axis 78. In the FRC, there aretwo regions of magnetic field lines: open 80 and closed 82. The surfacedividing the two regions is called the separatrix 84. The FRC forms acylindrical null surface 86 in which the magnetic field vanishes. In thecentral part 88 of the FRC the magnetic field does not changeappreciably in the axial direction. At the ends 90, the magnetic fielddoes change appreciably in the axial direction. The magnetic field alongthe center axis 78 reverses direction in the FRC, which gives rise tothe term “Reversed” in Field Reversed Configuration (FRC).

[0083] In FIG. 9A, the magnetic field outside of the null surface 94 isin the direction 96. The magnetic field inside the null surface is inthe direction 98. If an ion moves in the direction 100, the Lorentzforce 30 acting on it points towards the null surface 94. This is easilyappreciated by applying the right-hand rule. For particles moving in thedirection 102, called diamagnetic, the Lorentz force always pointstoward the null surface 94. This phenomenon gives rise to a particleorbit called betatron orbit, to be described below.

[0084]FIG. 9B shows an ion moving in the direction 104, calledcounterdiamagnetic. The Lorentz force in this case points away from thenull surface 94. This phenomenon gives rise to a type of orbit called adrift orbit, to be described below. The diamagnetic direction for ionsis counterdiamagnetic for electrons, and vice versa.

[0085]FIG. 10 shows a ring or annular layer of plasma 106 rotating inthe ions' diamagnetic direction 102. The ring 106 is located around thenull surface 86. The magnetic field 108 created by the annular plasmalayer 106, in combination with an externally applied magnetic field 110,forms a magnetic field having the topology of a FRC (The topology isshown in FIG. 8).

[0086] The ion beam that forms the plasma layer 106 has a temperature;therefore, the velocities of the ions form a Maxwell distribution in aframe rotating at the average angular velocity of the ion beam.Collisions between ions of different velocities lead to fusionreactions. For this reason, the plasma beam layer 106 is called acolliding beam system.

[0087]FIG. 11 shows the main type of ion orbits in a colliding beamsystem, called a betatron orbit 112. A betatron orbit 112 can beexpressed as a sine wave centered on the null circle 114. As explainedabove, the magnetic field on the null circle 114 vanishes. The plane ofthe orbit 112 is perpendicular to the axis 78 of the FRC. Ions in thisorbit 112 move in their diamagnetic direction 102 from a starting point116. An ion in a betatron orbit has two motions: an oscillation in theradial direction (perpendicular to the null circle 114), and atranslation along the null circle 114.

[0088]FIG. 12A is a graph of the magnetic field 118 in a FRC. The field118 is derived using a one-dimensional equilibrium model, to bediscussed below in conjunction with the theory of the invention. Thehorizontal axis of the graph represents the distance in centimeters fromthe FRC axis 78. The magnetic field is in kilogauss. As the graphdepicts, the magnetic field 118 vanishes at the null circle radius 120.

[0089] As shown in FIG. 12B, a particle moving near the null circle willsee a gradient 126 of the magnetic field pointing away from the nullsurface 86. The magnetic field outside the null circle is 122, while themagnetic field inside the null circle is 124. The direction of thegradient drift is given by the cross product {right arrow over (B)}×∇B,where ∇B is the gradient of the magnetic field; thus, it can beappreciated by applying the right-hand rule that the direction of thegradient drift is in the counterdiamagnetic direction, whether the ionis outside or inside the null circle 128.

[0090]FIG. 13A is a graph of the electric field 130 in a FRC. The field130 is derived using a one-dimensional equilibrium model, to bediscussed below in conjunction with the theory of the invention. Thehorizontal axis of the graph represents the distance in centimeters fromthe FRC axis 78. The electric field is in volts/cm. As the graphdepicts, the electric field 130 vanishes close to the null circle radius120.

[0091] As shown if FIG. 13B, the electric field for ions is deconfining;it points away from the null surface 132, 134. The magnetic field, asbefore, is in the directions 122,124. It can be appreciated by applyingthe right-hand rule that the direction of the {right arrow over(E)}×{right arrow over (B)} drift is in the diamagnetic direction,whether the ion is outside or inside the null surface 136.

[0092]FIGS. 14A and 14B show another type of common orbit in a FRC,called a drift orbit 138. Drift orbits 138 can be outside of the nullsurface, as shown in FIG. 14A, or inside it, as shown in FIG. 14B. Driftorbits 138 rotate in the diamagnetic direction if the {right arrow over(E)}×{right arrow over (B)} drift dominates or in the counterdiamagneticdirection if the gradient drift dominates. The drift orbits 138 shown inFIGS. 14A and 14B rotate in the diamagnetic direction 102 from startingpoint 116.

[0093] A drift orbit, as shown in FIG. 14C, can be thought of as a smallcircle rolling over a relatively bigger circle. The small circle 142spins around its axis in the sense 144. It also rolls over the bigcircle 146 in the direction 102. The point 140 will trace in space apath similar to 138.

[0094]FIGS. 15A and 15B show the direction of the Lorentz force at theends of a FRC. In FIG. 15A, an ion is shown moving in the diamagneticdirection 102 with a velocity 148 in a magnetic field 150. It can beappreciated by applying the right-hand rule that the Lorentz force 152tends to push the ion back into the region of closed field lines. Inthis case, therefore, the Lorentz force 152 is confining for the ions.In FIG. 15B, an ion is shown moving in the counterdiamagnetic directionwith a velocity 148 in a magnetic field 150. It can be appreciated byapplying the right-hand rule that the Lorentz force 152 tends to pushthe ion into the region of open field lines. In this case, therefore,the Lorentz force 152 is deconfining for the ions.

[0095] Magnetic and Electrostatic Confinement in a FRC

[0096] A plasma layer 106 (see FIG. 10) can be formed in a FRC byinjecting energetic ion beams around the null surface 86 in thediamagnetic direction 102 of ions. (A detailed discussion of differentmethods of forming the FRC and plasma ring follows below.) In thecirculating plasma layer 106, most of the ions have betatron orbits 112(see FIG. 11), are energetic, and are non-adiabatic; thus, they areinsensitive to short-wavelength fluctuations that cause anomaloustransport.

[0097] While studying a plasma layer 106 in equilibrium conditions asdescribed above, it was discovered that the conservation of momentumimposes a relation between the angular velocity of ions ω_(i) and theangular velocity of electrons ω_(e). (The derivation of this relation isgiven below in conjunction with the theory of the invention.) Therelation is $\begin{matrix}{{\omega_{e} = {\omega_{i}\left\lbrack {1 - \frac{\omega_{i}}{\Omega_{0}}} \right\rbrack}},{{{where}\quad \Omega_{0}} = {\frac{{Ze}\quad B_{0}}{m_{i}c}.}}} & (4)\end{matrix}$

[0098] In Eq. 4, Z is the ion atomic number, m_(i) is the ion mass, e isthe electron charge, B₀ is the magnitude of the applied magnetic field,and c is the speed of light. There are three free parameters in thisrelation: the applied magnetic field B₀, the electron angular velocityω_(e), and the ion angular velocity ω_(i). If two of them are known, thethird can be determined from Eq. 4.

[0099] Because the plasma layer 106 is formed by injecting ion beamsinto the FRC, the angular velocity of ions ω_(i) is determined by theinjection kinetic energy of the beam W_(i), which is given by$W_{i} = {{\frac{1}{2}m_{i}V_{i}^{2}} = {\frac{1}{2}{\left( {m_{i}\left( {\omega_{i}r_{o}} \right)} \right)^{2}.}}}$

[0100] Here, V_(i)=ω_(i)r₀, where V_(i) is the injection velocity ofions, ω_(i) is the cyclotron frequency of ions, and r₀ is the radius ofthe null surface 86. The kinetic energy of electrons in the beam hasbeen ignored because the electron mass m_(e) is much smaller than theion mass m_(i).

[0101] For a fixed injection velocity of the beam (fixed ω_(i)), theapplied magnetic field B₀ can be tuned so that different values of ω_(e)are obtainable. As will be shown, tuning the external magnetic field B₀also gives rise to different values of the electrostatic field insidethe plasma layer. This feature of the invention is illustrated in FIGS.16A and 16B. FIG. 16A shows three plots of the electric field (involts/cm) obtained for the same injection velocity, ω_(i)=1.35×10⁷s⁻¹,but for three different values of the applied magnetic field B₀: PlotApplied magnetic field (B₀) electron angular velocity (ω_(e)) 154 B₀ =2.77 kG ω_(e) = 0 156 B₀ = 5.15 kG ω_(e) = 0.625 × 10⁷ s⁻¹ 158 B₀ = 15.5kG ω_(e) = 1.11 × 10⁷ s⁻¹

[0102] The values of ω_(e) in the table above were determined accordingto Eq. 4. One can appreciate that ω_(e)>0 means that Ω₀>ω_(i) in Eq. 4,so that electrons rotate in their counterdiamagnetic direction. FIG. 16Bshows the electric potential (in volts) for the same set of values of B₀and ω_(e). The horizontal axis, in FIGS. 16A and 16B, represents thedistance from the FRC axis 78, shown in the graph in centimeters. Theanalytic expressions of the electric field and the electric potentialare given below in conjunction with the theory of the invention. Theseexpressions depend strongly on ω_(e).

[0103] The above results can be explained on simple physical grounds.When the ions rotate in the diamagnetic direction, the ions are confinedmagnetically by the Lorentz force. This was shown in FIG. 9A. Forelectrons, rotating in the same direction as the ions, the Lorentz forceis in the opposite direction, so that electrons would not be confined.The electrons leave the plasma and, as a result, a surplus of positivecharge is created. This sets up an electric field that prevents otherelectrons from leaving the plasma. The direction and the magnitude ofthis electric field, in equilibrium, is determined by the conservationof momentum. The relevant mathematical details are given below inconjunction with the theory of the invention.

[0104] The electrostatic field plays an essential role on the transportof both electrons and ions. Accordingly, an important aspect of thisinvention is that a strong electrostatic field is created inside theplasma layer 106, the magnitude of this electrostatic field iscontrolled by the value of the applied magnetic field B₀ which can beeasily adjusted.

[0105] As explained, the electrostatic field is confining for electronsif ω_(e)>0. As shown in FIG. 16B, the depth of the well can be increasedby tuning the applied magnetic field B₀. Except for a very narrow regionnear the null circle, the electrons always have a small gyroradius.Therefore, electrons respond to short-wavelength fluctuations with ananomalously fast diffusion rate. This diffusion, in fact, helps maintainthe potential well once the fusion reaction occurs. The fusion productions, being of much higher energy, leave the plasma. To maintain chargequasi-neutrality, the fusion products must pull electrons out of theplasma with them, mainly taking the electrons from the surface of theplasma layer. The density of electrons at the surface of the plasma isvery low, and the electrons that leave the plasma with the fusionproducts must be replaced; otherwise, the potential well woulddisappear.

[0106]FIG. 17 shows a Maxwellian distribution 162 of electrons. Onlyvery energetic electrons from the tail 160 of the Maxwell distributioncan reach the surface of the plasma and leave with fusion ions. The tail160 of the distribution 162 is thus continuously created byelectron-electron collisions in the region of high density near the nullsurface. The energetic electrons still have a small gyroradius, so thatanomalous diffusion permits them to reach the surface fast enough toaccommodate the departing fusion product ions. The energetic electronslose their energy ascending the potential well and leave with verylittle energy. Although the electrons can cross the magnetic fieldrapidly, due to anomalous transport, anomalous energy losses tend to beavoided because little energy is transported.

[0107] Another consequence of the potential well is a strong coolingmechanism for electrons that is similar to evaporative cooling. Forexample, for water to evaporate, it must be supplied the latent heat ofvaporization. This heat is supplied by the remaining liquid water andthe surrounding medium, which then thermalize rapidly to a lowertemperature faster than the heat transport processes can replace theenergy. Similarly, for electrons, the potential well depth is equivalentto water's latent heat of vaporization. The electrons supply the energyrequired to ascend the potential well by the thermalization process thatre-supplies the energy of the Maxwell tail so that the electrons canescape. The thermalization process thus results in a lower electrontemperature, as it is much faster than any heating process. Because ofthe mass difference between electrons and protons, the energy transfertime from protons is about 1800 times less than the electronthermalization time. This cooling mechanism also reduces the radiationloss of electrons. This is particularly important for advanced fuels,where radiation losses are enhanced by fuel ions with atomic number Z>1.

[0108] The electrostatic field also affects ion transport. The majorityof particle orbits in the plasma layer 106 are betatron orbits 112.Large-angle collisions, that is, collisions with scattering anglesbetween 90° and 180°, can change a betatron orbit to a drift orbit. Asdescribed above, the direction of rotation of the drift orbit isdetermined by a competition between the {right arrow over (E)}×{rightarrow over (B)} drift and the gradient drift. If the {right arrow over(E)}×{right arrow over (B)} drift dominates, the drift orbit rotates inthe diamagnetic direction. If the gradient drift dominates, the driftorbit rotates in the counterdiamagnetic direction. This is shown inFIGS. 18A and 18B. FIG. 18A shows a transition from a betatron orbit toa drift orbit due to a 180° collision, which occurs at the point 172.The drift orbit continues to rotate in the diamagnetic direction becausethe {right arrow over (E)}×{right arrow over (B)} drift dominates. FIG.18B shows another 180° collision, but in this case the electrostaticfield is weak and the gradient drift dominates. The drift orbit thusrotates in the counterdiamagnetic direction.

[0109] The direction of rotation of the drift orbit determines whetherit is confined or not. A particle moving in a drift orbit will also havea velocity parallel to the FRC axis. The time it takes the particle togo from one end of the FRC to the other, as a result of its parallelmotion, is called transit time; thus, the drift orbits reach an end ofthe FRC in a time of the order of the transit time. As shown inconnection with FIG. 15A, the Lorentz force at the ends is confiningonly for drift orbits rotating in the diamagnetic direction. After atransit time, therefore, ions in drift orbits rotating in thecounterdiamagnetic direction are lost.

[0110] This phenomenon accounts for a loss mechanism for ions, which isexpected to have existed in all FRC experiments. In fact, in theseexperiments, the ions carried half of the current and the electronscarried the other half. In these conditions the electric field insidethe plasma was negligible, and the gradient drift always dominated the{right arrow over (E)}×{right arrow over (B)} drift. Hence, all thedrift orbits produced by large-angle collisions were lost after atransit time. These experiments reported ion diffusion rates that werefaster than those predicted by classical diffusion estimates.

[0111] If there is a strong electrostatic field, the {right arrow over(E)}×{right arrow over (B)} drift dominates the gradient drift, and thedrift orbits rotate in the diamagnetic direction. This was shown abovein connection with FIG. 18A. When these orbits reach the ends of theFRC, they are reflected back into the region of closed field lines bythe Lorentz force; thus, they remain confined in the system.

[0112] The electrostatic fields in the colliding beam system may bestrong enough, so that the {right arrow over (E)}×{right arrow over (B)}drift dominates the gradient drift. Thus, the electrostatic field of thesystem would avoid ion transport by eliminating this ion loss mechanism,which is similar to a loss cone in a mirror device.

[0113] Another aspect of ion diffusion can be appreciated by consideringthe effect of small-angle, electron-ion collisions on betatron orbits.FIG. 19A shows a betatron orbit 112; FIG. 19B shows the same orbit 112when small-angle electron-ion collisions are considered 174; FIG. 19Cshows the orbit of FIG. 19B followed for a time that is longer by afactor often 176; and FIG. 19D shows the orbit of FIG. 19B followed fora time longer by a factor of twenty 178. It can be seen that thetopology of betatron orbits does not change due to small-angle,electron-ion collisions; however, the amplitude of their radialoscillations grows with time. In fact, the orbits shown in FIGS. 19A to19D fatten out with time, which indicates classical diffusion.

[0114] Theory of the Invention

[0115] For the purpose of modeling the invention, a one-dimensionalequilibrium model for the colliding beam system is used, as shown inFIG. 10. The results described above were drawn from this model. Thismodel shows how to derive equilibrium expressions for the particledensities, the magnetic field, the electric field, and the electricpotential. The equilibrium model presented herein is valid for a plasmafuel with one type of ions (e.g., in a D-D reaction) or multiple typesof ions (e.g., D-T, D-He³, and p-B¹¹).

[0116] Vlasov-Maxwell Equations

[0117] Equilibrium solutions for the particle density and theelectromagnetic fields in a FRC are obtained by solvingself-consistently the Vlasov-Maxwell equations: $\begin{matrix}{{\frac{\partial f_{j}}{\partial t} + {\left( {\overset{\rightarrow}{v} \cdot \nabla} \right)f_{j}} + {{\frac{e_{j}}{m_{j}}\left\lbrack {\overset{\rightarrow}{E} + {\frac{\overset{\rightarrow}{v}}{c} \times \overset{\rightarrow}{B}}} \right\rbrack} \cdot {\nabla_{v}f_{j}}}} = 0} & (5) \\{{\nabla{\times \overset{\rightarrow}{E}}} = {{- \frac{1}{c}}\frac{\partial\overset{\rightarrow}{B}}{\partial t}}} & (6) \\{{\nabla{\times \overset{\rightarrow}{B}}} = {{\frac{4\pi}{c}{\sum\limits_{j}{e_{j}{\int{\overset{\rightarrow}{v}f_{j}{\overset{\rightarrow}{v}}}}}}} + {\frac{1}{c}\frac{\partial\overset{\rightarrow}{E}}{\partial t}}}} & (7) \\{{\nabla{\cdot \overset{\rightarrow}{E}}} = {4\pi {\sum\limits_{j}{e_{j}{\int{f_{j}{\overset{\rightarrow}{v}}}}}}}} & (8)\end{matrix}$

 ∇·{right arrow over (B)}=0,  (9)

[0118] where j=e, i and i=1, 2, . . . for electrons and each species ofions. In equilibrium, all physical quantities are independent of time(i.e., ∂/∂t=0). To solve the Vlasov-Maxwell equations, the followingassumptions and approximations are made:

[0119] (a) All the equilibrium properties are independent of axialposition z (i.e., ∂/∂z=0). This corresponds to considering a plasma withan infinite extension in the axial direction; thus, the model is validonly for the central part 88 of a FRC.

[0120] (b) The system has cylindrical symmetry. Hence, all equilibriumproperties do not depend on θ(i.e., ∂/∂θ=0).

[0121] (c) The Gauss law, Eq. 8, is replaced with the quasi-neutralitycondition: Σ_(j)n_(j)e_(j)=0. By assuming infinite axial extent of theFRC and cylindrical symmetry, all the equilibrium properties will dependonly on the radial coordinate r. For this reason, the equilibrium modeldiscussed herein is called one-dimensional. With these assumptions andapproximations, the Vlasov-Maxwell equations reduce to: $\begin{matrix}{{{\left( {\overset{\rightarrow}{v} \cdot \nabla} \right)f_{j}} + {\frac{e_{j}}{m_{j}}{\overset{\rightarrow}{E} \cdot {\nabla_{v}f_{j}}}} + {{\frac{e_{j}}{m_{j}c}\left\lbrack {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}} \right\rbrack} \cdot {\nabla_{v}f_{j}}}} = 0} & (10) \\{{\nabla{\times \overset{\rightarrow}{B}}} = {\frac{4\pi}{c}{\sum\limits_{j}{e_{j}{\int{\overset{\rightarrow}{v}f_{j}{\overset{\rightarrow}{v}}}}}}}} & (11) \\{{\sum\limits_{\alpha}{n_{j}e_{j}}} = 0.} & (12)\end{matrix}$

[0122] Rigid Rotor Distributions

[0123] To solve Eqs. 10 through 12, distribution functions must bechosen that adequately describe the rotating beams of electrons and ionsin a FRC. A reasonable choice for this purpose are the so-called rigidrotor distributions, which are Maxwellian distributions in a uniformlyrotating frame of reference. Rigid rotor distributions are functions ofthe constants of motion: $\begin{matrix}{{{f_{j}\left( {r,\overset{\rightarrow}{v}} \right)} = {\left( \frac{m_{j}}{2\pi \quad T_{j}} \right)^{\frac{3}{2}}{n_{j}(0)}{\exp \left\lbrack {- \frac{ɛ_{j} - {\omega_{j}P_{j}}}{T_{j}}} \right\rbrack}}},} & (13)\end{matrix}$

[0124] where m_(j) is particle mass, {right arrow over (ν)} is velocity,T_(j) is temperature, n_(j)(0) is density at r=0, and ω_(j) is aconstant. The constants of the motion are$ɛ_{j} = {{\frac{m_{j}}{2}v^{2}} + {e_{j}\Phi \quad \left( {{for}\quad {energy}} \right)\quad {and}}}$${P_{j} = {{m_{j}\left( {{xv}_{y} - {yv}_{x}} \right)} + {\frac{e_{j}}{c}\Psi \quad \left( {{for}\quad {canonical}\quad {angular}\quad {momentum}} \right)}}},$

[0125] where Φ is the electrostatic potential and Ψ is the fluxfunction. The electromagnetic fields are$E_{r} = {{- \frac{\partial\Phi}{\partial r}}\quad \left( {{electric}\quad {field}} \right)\quad {and}}$$B_{z} = {\frac{1}{r}\frac{\partial\Psi}{\partial r}\quad {\left( {{magnetic}\quad {field}} \right).}}$

[0126] Substituting the expressions for energy and canonical angularmomentum into Eq. 13 yields $\begin{matrix}{{{f_{j}\left( {r,\overset{\rightarrow}{v}} \right)} = {\left( \frac{m_{j}}{2\pi \quad T_{j}} \right)^{\frac{3}{2}}{n_{j}(r)}\exp \left\{ {{- \frac{m_{j}}{2\quad T_{j}}}{{\overset{\rightarrow}{v} - {{\overset{\rightarrow}{\omega}}_{j} \times \overset{\rightarrow}{r}}}}^{2}} \right\}}},} & (14)\end{matrix}$

[0127] where

|{right arrow over (ν)}−{right arrow over (ω)}_(j)×{right arrow over(r)}|²=(ν_(x)+yω_(j))²+(ν_(y)−xω_(j))²+ν_(z) ²

[0128] and $\begin{matrix}{{n_{j}(r)} = {{n_{j}(0)}\exp {\left\{ {- {\frac{1}{T_{j}}\left\lbrack {{e_{j}\left( {\Phi - {\frac{\omega_{j}}{c}\Psi}} \right)} - {\frac{m_{j}}{2}\omega_{j}^{2}r^{2}}} \right\rbrack}} \right\}.}}} & (15)\end{matrix}$

[0129] That the mean velocity in Eq. 14 is a uniformly rotating vectorgives rise to the name rigid rotor. One of skill in the art canappreciate that the choice of rigid rotor distributions for describingelectrons and ions in a FRC is justified because the only solutions thatsatisfy Vlasov's equation (Eq. 10) are rigid rotor distributions (e.g.,Eq. 14). A proof of this assertion follows:

[0130] Proof

[0131] We require that the solution of Vlasov's equation (Eq. 10) be inthe form of a drifted Maxwellian: $\begin{matrix}{{{f_{j}\left( {\overset{\rightarrow}{r},\overset{\rightarrow}{v}} \right)} = {\left( \frac{m_{j}}{2\pi \quad {T_{j}(r)}} \right)^{\frac{3}{2}}{n_{j}(r)}{\exp \left\lbrack {{- \frac{m_{\alpha}}{2\quad {T_{j}(r)}}}\left( {\overset{\rightarrow}{v} - {{\overset{\rightarrow}{u}}_{j}(r)}} \right)^{2}} \right\rbrack}}},} & (16)\end{matrix}$

[0132] i.e., a Maxwellian with particle density n_(j)(r), temperatureT_(j)(r), and mean velocity u_(j)(r) that are arbitrary functions ofposition. Substituting Eq. 16 into the Vlasov's equation (Eq. 10) showsthat (a) the temperatures T_(j)(r) must be constants; (b) the meanvelocities {right arrow over (u)}_(j)(r) must be uniformly rotatingvectors; and (c) the particle densities n_(j)(r) must be of the form ofEq. 15. Substituting Eq. 16 into Eq. 10 yields a third-order polynomialequation in {right arrow over (ν)}:${\overset{\rightarrow}{v} \cdot {\nabla\left( {\ln \quad n_{j}} \right)}} + {{\frac{m_{j}\left( {\overset{\rightarrow}{v} - {\overset{\rightarrow}{u}}_{j}} \right)}{T_{j}} \cdot \left( {\overset{\rightarrow}{v} \cdot \nabla} \right)}{\overset{\rightarrow}{u}}_{j}} + {\frac{{m_{j}\left( {\overset{\rightarrow}{v} - {\overset{\rightarrow}{u}}_{j}} \right)}^{2}}{2T_{j}^{2}}\left( {\overset{\rightarrow}{v} \cdot \nabla} \right)T_{j}\ldots}$${\ldots + {\frac{e_{j}}{T_{j}}{\overset{\rightarrow}{E} \cdot \left( {\overset{\rightarrow}{v} - {\overset{\rightarrow}{u}}_{j}} \right)}} - {{\frac{e_{j}}{T_{j}c}\left\lbrack {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}} \right\rbrack} \cdot \left( {\overset{\rightarrow}{v} - {\overset{\rightarrow}{u}}_{j}} \right)}} = 0.$

[0133] Grouping terms of like order in {right arrow over (ν)} yields$\frac{m_{j}}{2\quad T_{j}^{2}}{\overset{\rightarrow}{v}}^{2}\left( {\overset{\rightarrow}{v} \cdot {\nabla T_{j}}} \right)\ldots$$\ldots + {\frac{m_{j}}{T_{j}}\left( {\overset{\rightarrow}{v} \cdot {\nabla{\overset{\rightarrow}{u}}_{j}} \cdot \overset{\rightarrow}{v}} \right)} - {\frac{m_{j}}{T_{j}^{2}}\left( {\overset{\rightarrow}{v} \cdot {\overset{\rightarrow}{u}}_{j}} \right)\left( {\overset{\rightarrow}{v} \cdot {\nabla T_{j}}} \right)\ldots}$$\ldots + {\overset{\rightarrow}{v} \cdot {\nabla\left( {\ln \quad n_{j}} \right)}} + {\frac{m_{j}}{2T_{j}^{2}}{{\overset{\rightarrow}{u}}_{j}}^{2}\left( {\overset{\rightarrow}{v} \cdot {\nabla T_{j}}} \right)} - {\frac{m_{j}}{T_{j}}\left( {\overset{\rightarrow}{v} \cdot {\nabla{\overset{\rightarrow}{u}}_{j}} \cdot {\overset{\rightarrow}{u}}_{j}} \right)} - {\frac{e_{j}}{T_{j}}{\overset{\rightarrow}{v} \cdot \overset{\rightarrow}{E}}} + {\frac{e_{j}}{{cT}_{j}}{\left( {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}} \right) \cdot {\overset{\rightarrow}{u}}_{j}}\ldots}$${\ldots + {\frac{e_{j}}{T_{j}}{\overset{\rightarrow}{E} \cdot {\overset{\rightarrow}{u}}_{j}}}} = 0.$

[0134] For this polynomial equation to hold for all {right arrow over(ν)}, the coefficient of each power of {right arrow over (ν)} mustvanish.

[0135] The third-order equation yields T_(j)(r)=constant.

[0136] The second-order equation gives $\begin{matrix}{{\overset{\rightarrow}{v} \cdot {\nabla{\overset{\rightarrow}{u}}_{j}} \cdot \overset{\rightarrow}{v}} = \quad {\left( {v_{x}v_{y}v_{z}} \right)\begin{pmatrix}\frac{\partial u_{x}}{\partial x} & \frac{\partial u_{y}}{\partial x} & \frac{\partial u_{z}}{\partial x} \\\frac{\partial u_{x}}{\partial y} & \frac{\partial u_{y}}{\partial y} & \frac{\partial u_{z}}{\partial y} \\\frac{\partial u_{x}}{\partial z} & \frac{\partial u_{y}}{\partial z} & \frac{\partial u_{z}}{\partial z}\end{pmatrix}\begin{pmatrix}v_{x} \\v_{y} \\v_{z}\end{pmatrix}}} \\{= \quad {{v_{x}^{2}\frac{\partial u_{x}}{\partial x}} + {v_{y}^{2}\frac{\partial u_{y}}{\partial y}} + {v_{z}^{2}\frac{\partial u_{z}}{\partial z}} + {v_{x}{v_{y}\left( {\frac{\partial u_{y}}{\partial x} + \frac{\partial u_{x}}{\partial y}} \right)}\ldots}}} \\{\quad {{\ldots + {v_{x}{v_{z}\left( {\frac{\partial u_{z}}{\partial x} + \frac{\partial u_{x}}{\partial z}} \right)}} + {v_{y}{v_{z}\left( {\frac{\partial u_{z}}{\partial y} + \frac{\partial u_{y}}{\partial z}} \right)}}} = 0.}}\end{matrix}$

[0137] For this to hold for all {right arrow over (ν)}, we must satisfy${\frac{\partial u_{x}}{\partial x} = {\frac{\partial u_{y}}{\partial y} = {\frac{\partial u_{z}}{\partial z} = {{0\quad {{and}\text{}\left( {\frac{\partial u_{y}}{\partial x} + \frac{\partial u_{x}}{\partial y}} \right)}} = {\left( {\frac{\partial u_{z}}{\partial x} + \frac{\partial u_{x}}{\partial z}} \right) = {\left( {\frac{\partial u_{z}}{\partial y} + \frac{\partial u_{y}}{\partial z}} \right) = 0}}}}}},$

[0138] which is solved generally by

{right arrow over (u)}_(j)({right arrow over (r)})=({right arrow over(ω)}_(j)×{right arrow over (r)})+{right arrow over (u)}_(0j)  (17)

[0139] In cylindrical coordinates, take {right arrow over (u)}_(0j)=0and {right arrow over (ω)}_(j)=ω_(j){circumflex over (z)}, whichcorresponds to injection perpendicular to a magnetic field in the{circumflex over (z)} direction. Then, {right arrow over (u)}_(j)({rightarrow over (r)})=ω_(j)r{circumflex over (θ)}.

[0140] The zero order equation indicates that the electric field must bein the radial direction, i.e., {right arrow over (E)}=E_(r){circumflexover (r)}.

[0141] The first-order equation is now given by $\begin{matrix}{{{\overset{\rightarrow}{v} \cdot {\nabla\left( {\ln \quad n_{j}} \right)}} - {\frac{m_{j}}{T_{j}}\left( {\overset{\rightarrow}{v} \cdot {\nabla{\overset{\rightarrow}{u}}_{j}} \cdot {\overset{\rightarrow}{u}}_{j}} \right)} - {\frac{e_{j}}{T_{j}}{\overset{\rightarrow}{v} \cdot \overset{\rightarrow}{E}}} + {\frac{e_{j}}{c\quad T_{j}}{\left( {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}} \right) \cdot {\overset{\rightarrow}{u}}_{j}}}} = 0.} & (18)\end{matrix}$

[0142] The second term in Eq. 18 can be rewritten with $\begin{matrix}{{{\nabla{\overset{\rightarrow}{u}}_{j}} \cdot {\overset{\rightarrow}{u}}_{j}} = {{\begin{pmatrix}\frac{\partial u_{r}}{\partial r} & \frac{\partial u_{\theta}}{\partial r} & \frac{\partial u_{z}}{\partial r} \\{\frac{1}{r}\frac{\partial u_{r}}{\partial\theta}} & {\frac{1}{r}\frac{\partial u_{\theta}}{\partial\theta}} & {\frac{1}{r}\frac{\partial u_{z}}{\partial\theta}} \\\frac{\partial u_{r}}{\partial z} & \frac{\partial u_{\theta}}{\partial z} & \frac{\partial u_{z}}{\partial z}\end{pmatrix}\begin{pmatrix}u_{r} \\u_{\theta} \\u_{z}\end{pmatrix}} = {{\begin{pmatrix}0 & \omega_{j} & 0 \\0 & 0 & 0 \\0 & 0 & 0\end{pmatrix}\begin{pmatrix}0 \\{\omega_{j}r} \\0\end{pmatrix}} = {\omega_{j}^{2}r{\hat{r}.}}}}} & (19)\end{matrix}$

[0143] The fourth term in Eq. 18 can be rewritten with $\begin{matrix}\begin{matrix}{{\left( {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}} \right) \cdot {\overset{\rightarrow}{u}}_{j}} = \quad {\overset{\rightarrow}{v} \cdot \left( {\overset{\rightarrow}{B} \times {\overset{\rightarrow}{u}}_{j}} \right)}} \\{= \quad {\overset{\rightarrow}{v} \cdot \left( {\left( {\nabla{\times \overset{\rightarrow}{A}}} \right) \times {\overset{\rightarrow}{u}}_{j}} \right)}} \\{= \quad {\overset{\rightarrow}{v} \cdot \left\lbrack {\left( {\frac{1}{r}\frac{\partial}{\partial r}\left( {r\quad A_{\theta}} \right)\hat{z}} \right) \times \left( {{- \omega_{j}}r\quad \hat{\theta}} \right)} \right\rbrack}} \\{= \quad {{\overset{\rightarrow}{v} \cdot \omega_{j}}\frac{\partial}{\partial r}\left( {r\quad A_{\theta}} \right)\hat{r}}}\end{matrix} & (20)\end{matrix}$

[0144] Using Eqs. 19 and 20, the first-order Eq. 18 becomes${{\frac{\partial}{\partial r}\left( {\ln \quad n_{j}} \right)} - {\frac{m_{j}}{T_{j}}\omega_{j}^{2}r} - {\frac{e_{j}}{T_{j}}E_{r}} + {\frac{e_{j}\omega_{j}}{c\quad T_{j}}\frac{\partial}{\partial r}\left( {r\quad {A_{\theta}(r)}} \right)}} = 0.$

[0145] The solution of this equation is $\begin{matrix}{{{n_{j}(r)} = {{n_{j}(0)}{\exp \left\lbrack {\frac{m_{j}\omega_{j}^{2}r^{2}}{2T_{j}} - \frac{e_{j}{\Phi (r)}}{T_{j}} - \frac{e_{j}\omega_{j}r\quad {A_{\theta}(r)}}{c\quad T_{j}}} \right\rbrack}}},} & (21)\end{matrix}$

[0146] where E_(r)=−dΦ/dr and n_(j)(0) is given by $\begin{matrix}{{n_{j}(0)} = {n_{j0}{{\exp \left\lbrack {{- \frac{m_{j}\omega_{j}^{2}r_{0}^{2}}{2T_{j}}} + \frac{e_{j}{\Phi \left( r_{0} \right)}}{T_{j}} + \frac{e_{j}\omega_{j}r_{0}{A_{\theta}\left( r_{0} \right)}}{c\quad T_{j}}} \right\rbrack}.}}} & (22)\end{matrix}$

[0147] Here, n_(j0) is the peak density at r₀.

[0148] Solution of Vlasov-Maxwell Equations

[0149] Now that it has been proved that it is appropriate to describeions and electrons by rigid rotor distributions, the Vlasov's equation(Eq. 10) is replaced by its first-order moments, i.e., $\begin{matrix}{{{{- n_{j}}m_{j}r\quad \omega_{j}^{2}} = {{n_{j}{e_{j}\left\lbrack {E_{r} + {\frac{r\quad \omega_{j}}{c}B_{z}}} \right\rbrack}} - {T_{j}\frac{n_{j}}{r}}}},} & (23)\end{matrix}$

[0150] which are conservation of momentum equations. The system ofequations to obtain equilibrium solutions reduces to: $\begin{matrix}{{{{- n_{j}}m_{j}r\quad \omega_{j}^{2}} = {{{n_{j}{e_{j}\left\lbrack {E_{r} + {\frac{r\quad \omega_{j}}{c}B_{z}}} \right\rbrack}} - {T_{j}\frac{n_{j}}{r}\quad j}} = e}},{i = 1},2,\ldots} & (24) \\{{{- \frac{\partial}{\partial r}}\frac{1}{r}\frac{\partial\Psi}{\partial r}} = {{- \frac{\partial B_{z}}{\partial r}} = {{\frac{4\pi}{c}j_{\theta}} = {\frac{4\pi}{c}r{\sum\limits_{j}{n_{j}e_{j}\omega_{j}}}}}}} & (25) \\{{\sum\limits_{j}{n_{j}e_{j}}} \cong 0.} & (26)\end{matrix}$

[0151] Solution for Plasma with One Type of Ion

[0152] Consider first the case of one type of ion fully stripped. Theelectric charges are given by e_(j)=−e,Ze. Solving Eq. 24 for E_(r) withthe electron equation yields $\begin{matrix}{{E_{r} = {{\frac{m}{e}r\quad \omega_{e}^{2}} - {\frac{r\quad \omega_{e}}{c}B_{z}} - {\frac{T_{e}}{e\quad n_{e}}\frac{n_{e}}{r}}}},} & (27)\end{matrix}$

[0153] and eliminating E_(r) from the ion equation yields$\begin{matrix}{{\frac{1}{r}{\log}\quad \frac{n_{i}}{r}} = {{\frac{Z_{i}e}{c}\frac{\left( {\omega_{i} - \omega_{e}} \right)}{T_{i}}B_{z}} - {\frac{Z_{2}T_{e}}{T_{i}}\frac{1}{r}{\log}\quad \frac{n_{e}}{r}} + \frac{m_{i}\omega_{i}^{2}}{T_{i}} + {\frac{m\quad Z_{i}\omega_{e}^{2}}{T_{i}}.}}} & (28)\end{matrix}$

[0154] Differentiating Eq. 28 with respect to r and substituting Eq. 25for dB_(z)/dr yields${{- \frac{B_{z}}{r}} = {{\frac{4\pi}{c}n_{e}e\quad {r\left( {\omega_{i} - \omega_{e}} \right)}\quad {and}\quad Z_{i}n_{i}} = n_{e}}},$

[0155] with T_(e)=T_(i)=constant, and ω_(i), ω_(e), constants, obtaining$\begin{matrix}{{\frac{1}{r}\frac{}{r}\frac{1}{r}{\log}\quad \frac{n_{i}}{r}} = {{\frac{4\pi \quad n_{e}Z_{i}e^{2}}{T_{i}}\frac{\left( {\omega_{i} - \omega_{e}} \right)^{2}}{c^{2}}} - {\frac{Z_{i}T_{e}}{T_{i}}\frac{1}{r}\frac{}{r}\frac{1}{r}{\frac{{\log}\quad n_{e}}{r}.}}}} & (29)\end{matrix}$

[0156] The new variable ξ is introduced: $\begin{matrix}{\xi = {\left. \frac{r^{2}}{2r_{0}^{2}}\Rightarrow{\frac{1}{r}\frac{}{r}\frac{1}{r}\frac{}{r}} \right. = {\frac{1}{r_{0}^{4}}{\frac{d^{2}}{d^{2}\xi}.}}}} & (30)\end{matrix}$

[0157] Eq. 29 can be expressed in terms of the new variable ξ:$\begin{matrix}{\frac{d^{2}\log \quad n_{i}}{d^{2}\xi} = {{\frac{4\pi \quad n_{e}Z_{i}e^{2}r_{0}^{4}}{T_{i}}\frac{\left( {\omega_{i} - \omega_{e}} \right)^{2}}{c^{2}}} - {\frac{Z_{i}T_{e}}{T_{i}}{\frac{d^{2}\log \quad n_{e}}{d^{2}\xi}.}}}} & (31)\end{matrix}$

[0158] Using the quasi-neutrality condition, $\begin{matrix}{{{n_{e} = {\left. {Z_{i}n_{i}}\Rightarrow\frac{d^{2}\log \quad n_{e}}{d^{2}\xi} \right. = \frac{d^{2}\log \quad n_{i}}{d^{2}\xi}}},{yields}}{\frac{d^{2}\log \quad n_{i}}{d^{2}\xi} = {- \frac{r_{0}^{4}}{\frac{\left( {T_{i} + {Z_{i}T_{e}}} \right)}{4\pi \quad Z_{i}^{2}e^{2}}\frac{c^{2}}{\left( {\omega_{i} - \omega_{e}} \right)^{2}}}}}{n_{i} = {- \frac{r_{0}^{4}}{\frac{\left( {T_{e} + \frac{T_{i}}{Z_{i}}} \right)}{4\pi \quad n_{e0}e^{2}}\frac{c^{2}}{\left( {\omega_{i} - \omega_{e}} \right)^{2}}}}}{\frac{n_{i}}{n_{i0}} = {{- 8}\left( \frac{r_{0}}{\Delta \quad r} \right)^{2}{\frac{n_{i}}{n_{i0}}.}}}} & (32)\end{matrix}$

[0159] Here is defined $\begin{matrix}{{{r_{0}\Delta \quad r} \equiv {2\sqrt{2}\left\{ \frac{T_{e} + \frac{T_{i}}{Z_{i}}}{4\pi \quad n_{e0}e^{2}} \right\}^{\frac{1}{2}}\frac{c}{{\omega_{i} - \omega_{e}}}}},} & (33)\end{matrix}$

[0160] where the meaning of Δr will become apparent soon. IfN_(i)=n_(i)n_(i0), where n_(i0) is the peak density at r=r₀, Eq. 32becomes $\begin{matrix}{\frac{d^{2}\log \quad N_{i}}{d^{2}\xi} = {{- 8}\left( \frac{r_{0}}{\Delta \quad r} \right)^{2}{N_{i}.}}} & (34)\end{matrix}$

[0161] Using another new variable,${\chi = {2\frac{r_{0}}{\Delta \quad r}\xi}},{{{yields}\quad \frac{d^{2}N_{i}}{d^{2}\chi}} = {{- 2}N_{i}}},$

[0162] the solution to which is${N_{i} = \frac{1}{\cos \quad {h^{2}\left( {\chi - \chi_{0}} \right)}}},$

[0163] where χ₀=χ(r₀) because of the physical requirement thatN_(i)(r₀)=1.

[0164] Finally, the ion density is given by $\begin{matrix}\begin{matrix}{n_{i} = \quad \frac{n_{i0}}{\cos \quad h^{2}2\left( \frac{r_{0}}{\Delta \quad r} \right)\left( {\xi - \frac{1}{2}} \right)}} \\{= \quad {\frac{n_{i0}}{\cos \quad {h^{2}\left( \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta \quad r} \right)}}.}}\end{matrix} & (35)\end{matrix}$

[0165] The significance of r₀ is that it is the location of peakdensity. Note that n_(i)(0)=n_(i)({square root}{square root over(2)}r₀). With the ion density known, B_(z) can be calculated using Eq.11, and E_(r) can be calculated using Eq. 27.

[0166] The electric and magnetic potentials are

Φ=−∫_(r′=0) ^(r′=0)E_(r)(r′)dr′ and

[0167] $\begin{matrix}\begin{matrix}{\Phi = \quad {- {\int_{r^{\prime} = 0}^{r^{\prime} = r}{{E_{r}\left( r^{\prime} \right)}{r^{\prime}}\quad {and}}}}} \\{A_{\theta} = \quad {\frac{1}{r}{\int_{r^{\prime} = 0}^{r^{\prime} = r}{r^{\prime}{B_{z}\left( r^{\prime} \right)}{r^{\prime}}}}}} \\{\Psi = \quad {r\quad A_{\theta}\quad \left( {{flux}\quad {function}} \right)}}\end{matrix} & (36)\end{matrix}$

[0168] Taking r={square root}{square root over (2)}r₀ to be the radiusat the wall (a choice that will become evident when the expression forthe electric potential Φ(r) is derived, showing that at r={squareroot}{square root over (2)}r₀ the potential is zero, i.e., a conductingwall at ground), the line density is $\begin{matrix}\begin{matrix}{N_{e} = \quad {Z_{i}N_{i}}} \\{= \quad {\int_{r = 0}^{r = {\sqrt{2}r_{0}}}\frac{n_{e0}2\pi {r}}{\cos \quad {h^{2}\left( \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta \quad r} \right)}}}} \\{= \quad {2\pi \quad n_{e0}r_{0}\Delta \quad r\quad \tan \quad h\quad \frac{r_{0}}{\Delta \quad r}\ldots}} \\{\ldots \cong \quad {2\pi \quad n_{e0}r_{0}\Delta \quad r\quad \left( {{{because}\quad r_{0}}{\Delta \quad r}} \right)}}\end{matrix} & (37)\end{matrix}$

[0169] Thus, Δr represents an “effective thickness.” In other words, forthe purpose of line density, the plasma can be thought of asconcentrated at the null circle in a ring of thickness Δr with constantdensity n_(e0).

[0170] The magnetic field is $\begin{matrix}{{B_{z}(r)} = {{B_{z}(0)} - {\frac{4\pi}{c}{\int_{r^{\prime} = 0}^{r^{\prime} = r}{{r^{\prime}}n_{e}e\quad {{r^{\prime}\left( {\omega_{i} - \omega_{e}} \right)}.}}}}}} & (38)\end{matrix}$

[0171] The current due to the ion and electron beams is $\begin{matrix}{I_{\theta} = {{\int_{0}^{\sqrt{2}r_{0}}{j_{\theta}{r}}} = {{\frac{N_{e}{e\left( {\omega_{i} - \omega_{e}} \right)}}{2\pi}\quad j_{\theta}} = {n_{0}e\quad {{r\left( {\omega_{i} - \omega_{e}} \right)}.}}}}} & (39)\end{matrix}$

[0172] Using Eq. 39, the magnetic field can be written as$\begin{matrix}\begin{matrix}{{B_{z}(r)} = \quad {{B_{z}(0)} - {\frac{2\pi}{c}I_{\theta}} - {\frac{2\pi}{c}I_{\theta}\tan \quad h\quad \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta \quad r}}}} \\{= \quad {{- B_{0}} - {\frac{2\pi}{c}I_{\theta}\tan \quad h\quad {\frac{r^{2} - r_{0}^{2}}{r_{0}\Delta \quad r}.}}}} \\{{{In}\quad {{Eq}.\quad 40}},} \\{\quad {{B_{z}(0)} = {{- B_{0}} + {\frac{2\pi}{c}I_{\theta}\quad {and}}}}} \\{\quad {{B_{z}\left( {\sqrt{2}r_{0}} \right)} = {{- B_{0}} - {\frac{2\pi}{c}{I_{\theta}.}}}}}\end{matrix} & (40)\end{matrix}$

[0173] If the plasma current I₀ vanishes, the magnetic field isconstant, as expected.

[0174] These relations are illustrated in FIGS. 20A through 20C. FIG.20A shows the external magnetic field B₀ 180. FIG. 20B shows themagnetic field due to the ring of current 182, the magnetic field havinga magnitude of (2π/c)I_(θ). FIG. 20C shows field reversal 184 due to theoverlapping of the two magnetic fields 180, 182.

[0175] The magnetic field is $\begin{matrix}\begin{matrix}{{B_{z}(r)} = \quad {- {B_{0}\left\lbrack {1 + {\frac{2\pi \quad I_{\theta}}{c\quad B_{0}}\tan \quad h\quad \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta \quad r}}} \right\rbrack}}} \\{{= \quad {- {B_{0}\left\lbrack {1 + {\sqrt{\beta}\tan \quad {h\left( \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta \quad r} \right)}}} \right\rbrack}}},}\end{matrix} & (41)\end{matrix}$

[0176] using the following definition for β: $\begin{matrix}{{{\frac{2\pi}{c}\frac{I_{\theta}}{B_{0}}} = {\frac{N_{e}{e\left( {\omega_{i} - \omega_{e}} \right)}}{c\quad B_{0}} = {\frac{2\pi}{c}n_{e0}\frac{r_{0}\Delta \quad {{re}\left( {\omega_{i} - \omega_{e}} \right)}}{B_{0}}\ldots}}}{\ldots = {\frac{2\pi}{c}2{\sqrt{2}\left\lbrack \frac{T_{e} + \left( {T_{i}/Z_{i}} \right)}{4\pi \quad n_{e0}e^{2}} \right\rbrack}^{\frac{1}{2}}\frac{c\quad n_{e0}}{\omega_{i} - \omega_{e}}\frac{e\left( {\omega_{i} - \omega_{e}} \right)}{B_{0}}\ldots}}{\ldots = {\left\lbrack \frac{8{\pi \left( {{n_{e0}T_{e}} + {n_{i0}T_{i}}} \right)}}{B_{0}^{2}} \right\rbrack^{\frac{1}{2}} \equiv {\sqrt{\beta}.}}}} & (42)\end{matrix}$

[0177] With an expression for the magnetic field, the electric potentialand the magnetic flux can be calculated. From Eq. 27, $\begin{matrix}{E_{r} = {{{{- \frac{r\quad \omega_{e}}{c}}B_{z}} - {\frac{T_{e}}{e}{\ln}\quad \frac{n_{e}}{r}} + {\frac{m}{e}r\quad \omega_{e}^{2}}} = {- \frac{\Phi}{r}}}} & (43)\end{matrix}$

[0178] Integrating both sides of Eq. 28 with respect to r and using thedefinitions of electric potential and flux function, $\begin{matrix}{{\Phi \equiv {- {\int_{r^{\prime} = 0}^{r^{\prime} = r}{E_{r}{r^{\prime}}\quad {and}\quad \Psi}}} \equiv {\int_{r^{\prime} = 0}^{r^{\prime} = r}{{B_{z}\left( r^{\prime} \right)}r^{\prime}{r^{\prime}}}}},{{which}\quad {yields}}} & (44) \\{\Phi = {{\frac{\omega_{e}}{e}\Psi} + {\frac{T_{e}}{e}\ln \quad \frac{n_{e}(r)}{n_{e}(0)}} - {\frac{m}{e}{\frac{r^{2}\omega_{e}^{2}}{2}.}}}} & (45)\end{matrix}$

[0179] Now, the magnetic flux can be calculated directly from theexpression of the magnetic field (Eq. 41): $\begin{matrix}{{\Psi = {\int_{r^{\prime} = 0}^{r^{\prime} = r}{{- {B_{0}\left\lbrack {1 + {\sqrt{\beta}\tan \quad h\quad \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta \quad r}}} \right\rbrack}}r^{\prime}{r^{\prime}}\ldots}}}{\ldots = {{- \frac{B_{o}r^{2}}{2}} - {\frac{B_{0}\sqrt{\beta}}{2}r_{0}\Delta \quad {r\left\lbrack {{\log \left( {\cos \quad h\quad \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta \quad r}} \right)} - {\log \left( {\cos \quad h\quad \frac{r_{o}}{\Delta \quad r}} \right)}} \right\rbrack}\ldots}}}{\ldots = {{- \frac{B_{0}r^{2}}{2}} + {B_{0}\frac{\sqrt{\beta}r_{0}\Delta \quad r}{4}\log \quad {\frac{n_{e}(r)}{n_{e}(0)}.}}}}} & (46)\end{matrix}$

[0180] Substituting Eq. 46 into Eq. 45 yields $\begin{matrix}{\Phi = {{\frac{\omega_{e}}{c}\frac{B_{0}\sqrt{\beta}r_{0}\Delta \quad r}{4}\log \quad \frac{n_{e}(r)}{n_{e}(0)}} + {\frac{T_{e}}{e}\ln \quad \frac{n_{e}(r)}{n_{e}(0)}} - {\frac{\omega_{e}}{c}B_{0}\frac{r^{2}}{2}} - {\frac{m}{e}{\frac{r^{2}\omega_{e}^{2}}{2}.}}}} & (47)\end{matrix}$

[0181] Using the definition of β, $\begin{matrix}{{{\frac{\omega_{e}}{c}B_{0}\sqrt{\beta}r_{0}\Delta \quad r} = {\frac{\omega_{e}}{c}\sqrt{8{\pi \left( {{n_{e0}T_{e}} + {n_{i0}T_{i}}} \right)}}2\frac{\left( {T_{e} + {T_{i}/2}} \right)^{\frac{1}{2}}}{\sqrt{4\pi \quad n_{e0}e^{2}}}\frac{c}{\left( {\omega_{i} - \omega_{e}} \right)}\ldots}}{\ldots = {4\frac{\omega_{e}}{\omega_{i} - \omega_{e}}{\frac{\left( {{n_{e0}T_{e}} + {n_{i0}T_{i}}} \right)}{n_{e0}e}.}}}} & (48)\end{matrix}$

[0182] Finally, using Eq. 48, the expressions for the electric potentialand the flux function become $\begin{matrix}{{\Psi (r)} = {{- \frac{B_{0}r^{2}}{2}} + {\frac{c}{\omega_{i} - \omega_{e}}\left( \frac{{n_{e0}T_{e}} + {n_{i0}T_{i}}}{n_{e0}e} \right)\ln \quad \frac{n_{e}(r)}{n_{e}(0)}\quad {and}}}} & (49) \\{{\Phi (r)} = {{\left\lbrack {{\frac{\omega_{e}}{\omega_{i} - \omega_{e}}\frac{\left( {{n_{e0}T_{e}} + {n_{i0}T_{i}}} \right)}{n_{e0}e}} + \frac{T_{e}}{e}} \right\rbrack \ln \quad \frac{n_{e}(r)}{n_{e}(0)}} - {\frac{\omega_{e}}{c}B_{0}\frac{r^{2}}{2}} - {\frac{m}{e}{\frac{r^{2}\omega_{e}^{2}}{c}.}}}} & (50)\end{matrix}$

[0183] Relationship Between ω_(i) and ω_(e)

[0184] An expression for the electron angular velocity ω_(i) can also bederived from Eqs. 24 through 26. It is assumed that ions have an averageenergy ½m_(i)(rω_(i))², which is determined by the method of formationof the FRC. Therefore, ω_(i) is determined by the FRC formation method,and ω_(e) can be determined by Eq. 24 by combining the equations forelectrons and ions to eliminate the electric field: $\begin{matrix}{{- \left\lbrack {{n_{e}m\quad r\quad \omega_{e}^{2}} + {n_{i}m_{i}r\quad \omega_{i}^{2}}} \right\rbrack} = {{\frac{n_{e}e\quad r}{c}\left( {\omega_{i} - \omega_{e}} \right)B_{z}} - {T_{e}\frac{n_{e}}{r}} - {T_{i}{\frac{n_{i}}{r}.}}}} & (51)\end{matrix}$

[0185] Eq. 25 can then be used to eliminate (ω_(i)-ω_(e)) to obtain$\begin{matrix}{\left\lbrack {{n_{e}m\quad r\quad \omega_{e}^{2}} + {n_{i}m_{i}r\quad \omega_{i}^{2}}} \right\rbrack = {\frac{}{r}{\left( {\frac{B_{z}^{2}}{8\pi} + {\sum\limits_{j}{n_{j}T_{j}}}} \right).}}} & (52)\end{matrix}$

[0186] Eq. 52 can be integrated from r=0 to r_(B)={square root over(2)}r₀. Assuming r₀/Δr>>1, the density is very small at both boundariesand B_(z)=−B₀ (1±{square root}{square root over (β)}). Carrying out theintegration shows $\begin{matrix}{{\left\lbrack {{n_{e0}m\quad \omega_{e}^{2}} + {n_{i0}m_{i}\omega_{i}^{2}}} \right\rbrack r_{0}\Delta \quad r} = {{\frac{B_{0}}{2\pi}\left\lbrack {8{\pi \left( {{n_{e0}T_{e}} + {n_{i0}T_{i}}} \right)}} \right\rbrack}^{\frac{1}{2}}.}} & (53)\end{matrix}$

[0187] Using Eq. 33 for Δr yields an equation for ω_(e): $\begin{matrix}{{{\omega_{i}^{2} + {\frac{Zm}{m_{i}}\omega_{e}^{2}}} = {\Omega_{0}\left( {\omega_{i} - \omega_{e}} \right)}},{{{where}\quad \Omega_{0}} = {\frac{{ZeB}_{0}}{m_{i}c}.}}} & (54)\end{matrix}$

[0188] Some limiting cases derived from Eq. 54 are:${{1.\quad \omega_{i}} = {{0\quad {and}\quad \omega_{e}} = {- \frac{{eB}_{0}}{mc}}}};$

 2. ω_(e)=0 and ω_(i)=Ω₀; and

[0189]${3.\quad \frac{Z\quad m}{m_{i}}\omega_{e}^{2}}{\omega_{i}^{2}\quad {and}\quad \omega_{e}} \cong {{\omega_{i}\left( {1 - \frac{\omega_{i}}{\Omega_{0}}} \right)}.}$

[0190] In the first case, the current is carried entirely by electronsmoving in their diamagnetic direction (ω_(e)<0). The electrons areconfined magnetically, and the ions are confined electrostatically by$\begin{matrix}{E_{r} = {\frac{T_{i}}{{Zen}_{i}}\frac{n_{i}}{r}\quad {\begin{matrix}{\leq {0\quad {for}\quad r} \geq r_{0}} \\{\geq {0\quad {for}\quad r} \leq r_{0}}\end{matrix}.}}} & (55)\end{matrix}$

[0191] In the second case, the current is carried entirely by ionsmoving in their diamagnetic direction (ω_(i)>0). If ω_(i) is specifiedfrom the ion energy ½m_(i)(rω₁)², determined in the formation process,then ω_(e)=0 and Ω₀=ω_(e) identifies the value of B₀, the externallyapplied magnetic field. The ions are magnetically confined, andelectrons are electrostatically confined by $\begin{matrix}{E_{r} = {\frac{T_{e}}{{en}_{e}}\frac{n_{e}}{r}\quad {\begin{matrix}{\geq {0\quad {for}\quad r} \geq r_{0}} \\{\leq {0\quad {for}\quad r} \leq r_{0}}\end{matrix}.}}} & (56)\end{matrix}$

[0192] In the third case, ω_(e)>0 and Ω₀>ω_(i). Electrons move in theircounter diamagnetic direction and reduce the current density. From Eq.33, the width of the distribution n_(i)(r) is increased; however, thetotal current/unit length is $\begin{matrix}{{I_{\theta} = {{\int_{r = 0}^{r_{B}}{j_{\theta}{r}}} = {\frac{N_{e}}{2\pi}{e\left( {\omega_{i} - \omega_{e}} \right)}}}},{where}} & (57)\end{matrix}$

 N _(e)=∫_(r=0) ^(r) ^(_(B)) 2πrdrn _(e)=2πr ₀ Δrn _(e0).  (58)

[0193] Here, r_(B)={square root}{square root over (2)}r₀ andr₀Δr≡(ω_(i)−ω_(e))⁻¹ according to Eq. 33. The electron angular velocityω_(e) can be increased by tuning the applied magnetic field B₀. Thisdoes not change either I_(θ) or the maximum magnetic field produced bythe plasma current, which is B₀{square root}{square root over(β)}=(2π/c)I₀. However, it does change Δr and, significantly, thepotential Φ. The maximum value of Φ is increased, as is the electricfield that confines the electrons.

[0194] Tuning the Magnetic Field

[0195] In FIGS. 21A-D, the quantities n_(e)/n_(e0) 186, B_(z)/(B₀{squareroot}{square root over (β)}) 188, (Φ/Φ₀ 190, and Ψ/Ψ₀ 192 are plottedagainst r/r₀ 194 for various values of B₀. The values of potential andflux are normalized to (Φ₀=20(T_(e)+T_(i))/e and Ψ₀=(c/ω_(i))Φ₀. Adeuterium plasma is assumed with the following data: n_(e0)=n_(i0)=10¹⁵cm⁻³; r₀=40 cm; ½m_(i)(r₀ω_(i))²=300 keV; and T_(e)=T_(i)=100 keV. Foreach of the cases illustrated in FIG. 21, ω_(i)=1.35×10⁷ s⁻¹, and ω_(e)is determined from Eq. 54 for various values of B₀: Plot appliedmagnetic field (B₀) electron angular velocity (ω_(e)) 154 B₀ = 2.77 kGω_(e) = 0 156 B₀ = 5.15 kG ω_(e) = 0.625 × 10⁷ s⁻¹ 158 B₀ = 15.5 kGω_(e) = 1.11 × 10⁷ s⁻¹

[0196] The case of ω_(e)=−ω_(i) and B₀=1.385 kG involves magneticconfinement of both electrons and ions. The potential reduces toΦ/Φ₀=m_(i)(rω_(i))²/[80(T_(e)+T_(i))], which is negligible compared tothe case ω₃=0. The width of the density distribution Δr is reduced by afactor of 2, and the maximum magnetic field B₀{square root}{square rootover (β)} is the same as for ω_(e)=0.

[0197] Solution for Plasmas of Multiple Types of Ions

[0198] This analysis can be carried out to include plasmas comprisingmultiple types of ions. Fusion fuels of interest involve two differentkinds of ions, e.g., D-T, D-He³, and H-B¹¹. The equilibrium equations(Eqs. 24 through 26) apply, except that j=e, 1, 2 denotes electrons andtwo types of ions where Z₁=1 in each case and Z₂=Z=1, 2, 5 for the abovefuels. The equations for electrons and two types of ions cannot besolved exactly in terms of elementary functions. Accordingly, aniterative method has been developed that begins with an approximatesolution.

[0199] The ions are assumed to have the same values of temperature andmean velocity V_(i)=rω_(i). Ion-ion collisions drive the distributionstoward this state, and the momentum transfer time for the ion-ioncollisions is shorter than for ion-electron collisions by a factor of anorder of 1000. By using an approximation, the problem with two types ofions can be reduced to a single ion problem. The momentum conservationequations for ions are $\begin{matrix}{{{- n_{1}}m_{1}r\quad \omega_{1}^{2}} = {{n_{1}{e\left\lbrack {E_{r} + {\frac{r\quad \omega_{1}}{c}B_{z}}} \right\rbrack}} - {T_{i}\frac{n_{i}}{r}\quad {and}}}} & (59) \\{{{- n_{2}}m_{2}r\quad \omega_{2}^{2}} = {{n_{2}{{Ze}\left\lbrack {E_{r} + {\frac{r\quad \omega_{2}}{c}B_{z}}} \right\rbrack}} - {T_{2}{\frac{n_{2}}{r}.}}}} & (60)\end{matrix}$

[0200] In the present case, T₁=T₂ and ω₁=ω₂. Adding these two equationsresults in $\begin{matrix}{{{{- n_{1}}{\langle m_{i}\rangle}\omega_{i}^{2}} = {{n_{i}{\langle Z\rangle}{e\left\lbrack {E_{r} + {\frac{r\quad \omega_{i}}{c}B_{z}}} \right\rbrack}} - {T_{i}\frac{n_{i}}{r}}}},} & (61)\end{matrix}$

[0201] where n_(i)=n₁=n₂; ω_(i)=ω₁=ω₂; T_(i)=T₁=T₂; n₁m₁+n₂m₂; andn_(i)(Z)=n_(i)+n₂Z.

[0202] The approximation is to assume that (m_(i)) and (z) are constantsobtained by replacing n_(i)(r) and n₂(r) by n₁₀ and n₂₀, the maximumvalues of the respective functions. The solution of this problem is nowthe same as the previous solution for the single ion type, except that(Z) replaces Z and (m_(i)) replaces m_(i). The values of n₁ and n₂ canbe obtained from n₁+n₂=n_(i) and n₁+Zn₂=n_(e)=(Z)n_(i). It can beappreciated that n₁ and n₂ have the same functional form.

[0203] Now the correct solution can be obtained by iterating theequations: $\begin{matrix}{\frac{{\log}\quad N_{1}}{\xi} = {{m_{1}r_{0}^{2}\Omega_{1}\frac{\left( {\omega_{i} - \omega_{e}} \right)}{T_{i}}\frac{B_{z}(\xi)}{B_{0}}} - {\frac{T_{e}}{T_{i}}{\log}\quad \frac{N_{e}}{\xi}} + {\frac{{m_{1}\left( {\omega_{i}r_{0}} \right)}^{2}}{T_{i}}\quad {and}}}} & (62) \\{{{\frac{{\log}\quad N_{2}}{\xi} = {{m_{2}r_{0}^{2}\Omega_{2}\frac{\left( {\omega_{i} - \omega_{e}} \right)}{T_{i}}\frac{B_{z}(\xi)}{B_{0}}} - {\frac{Z\quad T_{e}}{T_{i}}{\log}\quad \frac{N_{e}}{\xi}} + \frac{{m_{2}\left( {\omega_{i}r_{0}} \right)}^{2}}{T_{i}}}},{w\quad h\quad e\quad r\quad e}}{{N_{1} = \frac{n_{1}(r)}{n_{10}}},{N_{2} = \frac{n_{2}(r)}{n_{20}}},{\xi = \frac{r^{2}}{2r_{0}^{2}}},{\Omega_{1} = \frac{e\quad B_{0}}{m_{1}c}},{{a\quad n\quad d\quad \Omega_{2}} = {\frac{Z\quad e\quad B_{0}}{m_{2}c}.}}}} & (63)\end{matrix}$

[0204] The first iteration can be obtained by substituting theapproximate values of B_(z)(ξ) and N_(e)(ξ) in the right hand sides ofEqs. 62 and 63 and integrating to obtain the corrected values of n₁(r),n₂(r), and B_(z)(r).

[0205] Calculations have been carried out for the data shown in Table 1,below. Numerical results for fusion fuels are shown in FIGS. 22A-Dthrough 24A-D wherein the quantities n₁/n₁₀ 206, Φ/Φ₀ 208, and Ψ/Ψ₀ 210are plotted against r/r₀ 204. FIGS. 22A-D shows the first approximation(solid lines) and the final results (dotted lines) of the iterations forD-T for the normalized density of D 196, the normalized density of T198, the normalized electric potential 200, and the normalized flux 202.FIGS. 23A-D show the same iterations for D-He³ for the normalizeddensity of D 212, the normalized density of He³ 214, the normalizedelectric potential 216, and the normalized flux 218. FIGS. 24A-D showthe same iterations for p-B¹¹ for the normalized density of p 220, thenormalized density of B¹¹ 222, the normalized electric potential 224,and the normalized flux 226. Convergence of the iteration is most rapidfor D-T. In all cases the first approximation is close to the finalresult. TABLE 1 Numerical data for equilibrium calculations fordifferent fusion fuels Quantity Units D-T D-He³ p-B¹¹ n_(e0) cm⁻³ 10¹⁵10¹⁵ 10¹⁵ n₁₀ cm⁻³ 0.5 × 10¹⁵ ⅓ × 10¹⁵ 0.5 × 10¹⁵ n₂₀ cm⁻³ 0.5 × 10¹⁵ ⅓× 10¹⁵ 10¹⁴ v₁ = v₂ $\frac{cm}{\sec}$

0.54 × 10⁹ 0.661 × 10⁹ 0.764 × 10⁹ $\frac{1}{2}m_{1}v_{1}^{2}$

keV 300 450 300 $\frac{1}{2}m_{2}v_{2}^{2}$

keV 450 675 3300 ω_(i) = ω₁ = ω₂ rad/s 1.35 × 10⁷ 1.65 × 10⁷ 1.91 × 10⁷r₀ cm 40 40 40 B₀ kG 5.88 8.25 15.3 ⟨Z_(i)⟩

None 1 3/2 1.67 ⟨m_(i)⟩

m_(p) 5/2 5/2 2.67$\Omega_{0} = \frac{{\langle Z_{i}\rangle}{eB}_{0}}{{\langle m_{i}\rangle}c}$

rad/s 2.35 × 10⁷ 4.95 × 10⁷ 9.55 × 10⁷$\omega_{e} = {\omega_{i}\quad\left\lbrack {1 - \frac{\omega_{i}}{\Omega_{0}}} \right\rbrack}$

rad/s 0.575 × 10⁷ 1.1 × 10⁷ 1.52 × 10⁷ T_(e) keV 96 170 82 T_(i) keV 100217 235 r₀Δr cm² 114 203 313 β None 228 187 38.3

[0206] Structure of the Containment System

[0207]FIG. 25 illustrates a preferred embodiment of a containment system300 according to the present invention. The containment system 300comprises a chamber wall 305 that defines therein a confining chamber310. Preferably, the chamber 310 is cylindrical in shape, with principleaxis 315 along the center of the chamber 310. For application of thiscontainment system 300 to a fusion reactor, it is necessary to create avacuum or near vacuum inside the chamber 310. Concentric with theprinciple axis 315 is a betatron flux coil 320, located within thechamber 310. The betatron flux coil 320 comprises an electrical currentcarrying medium adapted to direct current around a long coil, as shown,which preferably comprises parallel winding multiple separate coils, andmost perferably parallel windings of about four separate coils, to forma long coil. Persons skilled in the art will appreciate that currentthrough the betatron coil 320 will result in a magnetic field inside thebetatron coil 320, substantially in the direction of the principle axis315.

[0208] Around the outside of the chamber wall 305 is an outer coil 325.The outer coil 325 produce a relatively constant magnetic field havingflux substantially parallel with principle axis 315. This magnetic fieldis azimuthally symmetrical. The approximation that the magnetic fielddue to the outer coil 325 is constant and parallel to axis 315 is mostvalid away from the ends of the chamber 310. At each end of the chamber310 is a mirror coil 330. The mirror coils 330 are adapted to produce anincreased magnetic field inside the chamber 310 at each end, thusbending the magnetic field lines inward at each end. (See FIGS. 8 and10.) As explained, this bending inward of the field lines helps tocontain the plasma 335 in a containment region within the chamber 310generally between the mirror coils 330 by pushing it away from the endswhere it can escape the containment system 300. The mirror coils 330 canbe adapted to produce an increased magnetic field at the ends by avariety of methods known in the art, including increasing the number ofwindings in the mirror coils 330, increasing the current through themirror coils 330, or overlapping the mirror coils 330 with the outercoil 325.

[0209] The outer coil 325 and mirror coils 330 are shown in FIG. 25implemented outside the chamber wall 305; however, they may be insidethe chamber 310. In cases where the chamber wall 305 is constructed of aconductive material such as metal, it may be advantageous to place thecoils 325, 330 inside the chamber wall 305 because the time that ittakes for the magnetic field to diffuse through the wall 305 may berelatively large and thus cause the system 300 to react sluggishly.Similarly, the chamber 310 may be of the shape of a hollow cylinder, thechamber wall 305 forming a long, annular ring. In such a case, thebetatron flux coil 320 could be implemented outside of the chamber wall305 in the center of that annular ring. Preferably, the inner wallforming the center of the annular ring may comprise a non-conductingmaterial such as glass. As will become apparent, the chamber 310 must beof sufficient size and shape to allow the circulating plasma beam orlayer 335 to rotate around the principle axis 315 at a given radius.

[0210] The chamber wall 305 may be formed of a material having a highmagnetic permeability, such as steel. In such a case, the chamber wall305, due to induced countercurrents in the material, helps to keep themagnetic flux from escaping the chamber 310, “compressing” it. If thechamber wall were to be made of a material having low magneticpermeability, such as plexiglass, another device for containing themagnetic flux would be necessary. In such a case, a series ofclosed-loop, flat metal rings could be provided. These rings, known inthe art as flux delimiters, would be provided within the outer coils 325but outside the circulating plasma beam 335. Further, these fluxdelimiters could be passive or active, wherein the active fluxdelimiters would be driven with a predetermined current to greaterfacilitate the containment of magnetic flux within the chamber 310.Alternatively, the outer coils 325 themselves could serve as fluxdelimiters.

[0211] As explained above, a circulating plasma beam 335, comprisingcharged particles, may be contained within the chamber 310 by theLorentz force caused by the magnetic field due to the outer coil 325. Assuch, the ions in the plasma beam 335 are magnetically contained inlarge betatron orbits about the flux lines from the outer coil 325,which are parallel to the principle axis 315. One or more beam injectionports 340 are also provided for adding plasma ions to the circulatingplasma beam 335 in the chamber 310. In a preferred embodiment, theinjector ports 340 are adapted to inject an ion beam at about the sameradial position from the principle axis 315 where the circulating plasmabeam 335 is contained (i.e., around the null surface). Further, theinjector ports 340 are adapted to inject ion beams 350 (See FIG. 28)tangent to and in the direction of the betatron orbit of the containedplasma beam 335.

[0212] Also provided are one or more background plasma sources 345 forinjecting a cloud of non-energetic plasma into the chamber 310. In apreferred embodiment, the background plasma sources 345 are adapted todirect plasma 335 toward the axial center of the chamber 310. It hasbeen found that directing the plasma this way helps to better containthe plasma 335 and leads to a higher density of plasma 335 in thecontainment region within the chamber 310.

[0213] Formation of the FRC

[0214] Conventional procedures used to form a FRC primarily employ thetheta pinch-field reversal procedure. In this conventional method, abias magnetic field is applied by external coils surrounding a neutralgas back-filled chamber. Once this has occurred, the gas is ionized andthe bias magnetic field is frozen in the plasma. Next, the current inthe external coils is rapidly reversed and the oppositely orientedmagnetic field lines connect with the previously frozen lines to formthe closed topology of the FRC (see FIG. 8). This formation process islargely empirical and there exists almost no means of controlling theformation of the FRC. The method has poor reproducibility and no tuningcapability as a result.

[0215] In contrast, the FRC formation methods of the present inventionallow for ample control and provide a much more transparent andreproducible process. In fact, the FRC formed by the methods of thepresent invention can be tuned and its shape as well as other propertiescan be directly influenced by manipulation of the magnetic field appliedby the outer field coils 325. Formation of the FRC by methods of thepresent inventions also results in the formation of the electric fieldand potential well in the manner described in detail above. Moreover,the present methods can be easily extended to accelerate the FRC toreactor level parameters and high-energy fuel currents, andadvantageously enables the classical confinement of the ions.Furthermore, the technique can be employed in a compact device and isvery robust as well as easy to implement—all highly desirablecharacteristics for reactor systems.

[0216] In the present methods, FRC formation relates to the circulatingplasma beam 335. It can be appreciated that the circulating plasma beam335, because it is a current, creates a poloidal magnetic field, aswould an electrical current in a circular wire. Inside the circulatingplasma beam 335, the magnetic self-field that it induces opposes theexternally applied magnetic field due to the outer coil 325. Outside theplasma beam 335, the magnetic self-field is in the same direction as theapplied magnetic field. When the plasma ion current is sufficientlylarge, the self-field overcomes the applied field, and the magneticfield reverses inside the circulating plasma beam 335, thereby formingthe FRC topology as shown in FIGS. 8 and 10.

[0217] The requirements for field reversal can be estimated with asimple model. Consider an electric current I_(p) carried by a ring ofmajor radius r₀ and minor radius a<<r₀. The magnetic field at the centerof the ring normal to the ring is B_(p)=2πI_(p)/(cr₀). Assume that thering current I_(p)=N_(p)e(Ω₀/2π) is carried by N_(p) ions that have anangular velocity Ω₀. For a single ion circulating at radiusr₀=V₀/Ω₀,Ω₀=eB₀/m_(i)c is the cyclotron frequency for an externalmagnetic field B₀. Assume V₀ is the average velocity of the beam ions.Field reversal is defined as $\begin{matrix}{{B_{p} = {\frac{N_{p}e\quad \Omega_{0}}{r_{0}c} \geq {2B_{0}}}},} & (64)\end{matrix}$

[0218] which implies that N_(p)>2 r₀/a_(i), and $\begin{matrix}{{I_{p} \geq \frac{e\quad V_{0}}{{\pi\alpha}_{i}}},} & (65)\end{matrix}$

[0219] where a₁=e²/m_(i)c²=1.57×10⁻¹⁶ cm and the ion beam energy is$\frac{1}{2}m_{i}{V_{0}^{2}.}$

[0220] In the one dimensional model, the magnetic field from the plasmacurrent is B_(p)=(2π/c)i_(p), where i_(p) is current per unit of length.The field reversal requirement is i_(p)>eV₀/πr₀α_(i)=0.225 kA/cm, whereB₀=69.3 G and ${\frac{1}{2}m_{i}V_{0}^{2}} = {100\quad {{eV}.}}$

[0221] For a model with periodic rings and B₀ is averaged over the axialcoordinate (B_(z))=(2π/c)(I_(p)/s) (s is the ring spacing), if s=r₀,this model would have the same average magnetic field as the onedimensional model with i_(p)=I_(p)/s.

[0222] Combined Beam/Betatron Formation Technique

[0223] A preferred method of forming a FRC within the confinement system300 described above is herein termed the combined beam/betatrontechnique. This approach combines low energy beams of plasma ions withbetatron acceleration using the betatron flux coil 320.

[0224] The first step in this method is to inject a substantiallyannular cloud layer of background plasma in the chamber 310 using thebackground plasma sources 345. Outer coil 325 produces a magnetic fieldinside the chamber 310, which magnetizes the background plasma. At shortintervals, low energy ion beams are injected into the chamber 310through the injector ports 340 substantially transverse to theexternally applied magnetic field within the chamber 310. As explainedabove, the ion beams are trapped within the chamber 310 in largebetatron orbits by this magnetic field. The ion beams may be generatedby an ion accelerator, such as an accelerator comprising an ion diodeand a Marx generator. (see R. B. Miller, An Introduction to the Physicsof Intense Charged Particle Beams, (1982)). As one of skill in the artcan appreciate, the externally applied magnetic field will exert aLorentz force on the injected ion beam as soon as it enters the chamber310; however, it is desired that the beam not deflect, and thus notenter a betatron orbit, until the ion beam reaches the circulatingplasma beam 335. To solve this problem, the ion beams are neutralizedwith electrons and directed through a substantially constantunidirectional magnetic field before entering the chamber 310. Asillustrated in FIG. 26, when the ion beam 350 is directed through anappropriate magnetic field, the positively charged ions and negativelycharged electrons separate. The ion beam 350 thus acquires an electricself-polarization due to the magnetic field. This magnetic field may beproduced by, e.g., a permanent magnet or by an electromagnet along thepath of the ion beam. When subsequently introduced into the confinementchamber 310, the resultant electric field balances the magnetic force onthe beam particles, allowing the ion beam to drift undeflected. FIG. 27shows a head-on view of the ion beam 350 as it contacts the plasma 335.As depicted, electrons from the plasma 335 travel along magnetic fieldlines into or out of the beam 350, which thereby drains the beam'selectric polarization. When the beam is no longer electricallypolarized, the beam joins the circulating plasma beam 335 in a betatronorbit around the principle axis 315, as shown in FIG. 25.

[0225] When the plasma beam 335 travels in its betatron orbit, themoving ions comprise a current, which in turn gives rise to a poloidalmagnetic self-field. To produce the FRC topology within the chamber 310,it is necessary to increase the velocity of the plasma beam 335, thusincreasing the magnitude of the magnetic self-field that the plasma beam335 causes. When the magnetic self-field is large enough, the directionof the magnetic field at radial distances from the axis 315 within theplasma beam 335 reverses, giving rise to a FRC. (See FIGS. 8 and 10). Itcan be appreciated that, to maintain the radial distance of thecirculating plasma beam 335 in the betatron orbit, it is necessary toincrease the applied magnetic field from the outer coil 325 as theplasma beam 335 increases in velocity. A control system is thus providedfor maintaining an appropriate applied magnetic field, dictated by thecurrent through the outer coil 325. Alternatively, a second outer coilmay be Hi used to provide the additional applied magnetic field that isrequired to maintain the radius of the plasma beam's orbit as it isaccelerated.

[0226] To increase the velocity of the circulating plasma beam 335 inits orbit, the betatron flux coil 320 is provided. Referring to FIG. 28,it can be appreciated that increasing a current through the betatronflux coil 320, by Ampere's Law, induces an azimuthal electric field, E,inside the chamber 310. The positively charged ions in the plasma beam335 are accelerated by this induced electric field, leading to fieldreversal as described above. When ion beams are added to the circulatingplasma beam 335, as described above, the plasma beam 335 depolarizes theion beams.

[0227] For field reversal, the circulating plasma beam 335 is preferablyaccelerated to a rotational energy of about 100 eV, and preferably in arange of about 75 eV to 125 eV. To reach fusion relevant conditions, thecirculating plasma beam 335 is preferably accelerated to about 200 keVand preferably to a range of about 100 keV to 3.3 MeV. In developing thenecessary expressions for the betatron acceleration, the acceleration ofsingle particles is first considered. The gyroradius of ions r=V/Ω_(i)will change because V increases and the applied magnetic field mustchange to maintain the radius of the plasma beam's orbit, r₀=V/Ω_(c)$\begin{matrix}{{\frac{\partial r}{\partial t} = {{\frac{1}{\Omega}\left\lbrack {\frac{\partial V}{\partial t} - {\frac{V}{\Omega_{i}}\frac{\partial\Omega_{i}}{\partial t}}} \right\rbrack} = 0}},{w\quad h\quad e\quad r\quad e}} & (66) \\{{\frac{\partial V}{\partial t} = {{\frac{r_{0}e}{m_{i}c}\frac{\partial B_{c}}{\partial t}} = {\frac{e\quad E_{\theta}}{m_{i}} = {{- \frac{e}{m_{i}c}}\frac{1}{2\pi \quad r_{0}}\frac{\partial\Psi}{\partial t}}}}},} & (67)\end{matrix}$

[0228] and Ψ is the magnetic flux: $\begin{matrix}{{\Psi = {{\int_{0}^{r_{0}}{B_{z}2\quad \pi \quad r{r}}} = {\pi \quad r_{0}^{2}{\langle B_{z}\rangle}}}},} & (68) \\{w\quad h\quad e\quad r\quad e} & \quad \\{{\langle B_{z}\rangle} = {{- {B_{F}\left( \frac{r_{a}}{r_{0}} \right)}^{2}} - {{B_{c}\left\lbrack {1 - \left( \frac{r_{a}}{r_{0}} \right)^{2}} \right\rbrack}.}}} & (69)\end{matrix}$

[0229] From Eq. 67, it follows that $\begin{matrix}{{\frac{\partial{\langle B_{z}\rangle}}{\partial t} = {{- 2}\frac{\partial B_{c}}{\partial t}}},} & (70)\end{matrix}$

[0230] and (B_(z))=−2B_(c)+B₀, assuming that the initial values of B_(F)and B_(c) are both B₀. Eq. 67 can be expressed as $\begin{matrix}{\frac{\partial V}{\partial t} = {{- \frac{e}{2m_{i}c}}r_{0}{\frac{\partial{\langle B_{z}\rangle}}{\partial t}.}}} & (71)\end{matrix}$

[0231] After integration from the initial to final states where${{\frac{1}{2}m\quad V_{0}^{2}} = {{W_{0}\quad a\quad n\quad d\quad \frac{1}{2}m\quad V^{2}} = W}},$

[0232] the final values of the magnetic fields are: $\begin{matrix}{{B_{c} = {{B_{0}\sqrt{\frac{W}{W_{0}}}} = {2.19\quad {kG}}}}{a\quad n\quad d}} & (72) \\{{B_{F} = {{B_{0}\left\lbrack {\sqrt{\frac{W}{W_{0}}} + {\left( \frac{r_{0}}{r_{a}} \right)^{2}\left( {\sqrt{\frac{W}{W_{0}}} - 1} \right)}} \right\rbrack} = {10.7\quad {kG}}}},} & (73)\end{matrix}$

[0233] assuming B₀=69.3 G, W/W₀=1000, and r₀/r_(a)=2. This calculationapplies to a collection of ions, provided that they are all located atnearly the same radius r₀ and the number of ions is insufficient toalter the magnetic fields.

[0234] The modifications of the basic betatron equations to accommodatethe present problem will be based on a one-dimensional equilibrium todescribe the multi-ring plasma beam, assuming the rings have spread outalong the field lines and the z-dependence can be neglected. Theequilibrium is a self-consistent solution of the Vlasov-Maxwellequations that can be summarized as follows:

[0235] (a) The density distribution is $\begin{matrix}{{n = \frac{n_{m}}{\cosh^{2}\left( \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta \quad r} \right)}},} & (74)\end{matrix}$

[0236] which applies to the electrons and protons (assuming quasineutrality); r₀ is the position of the density maximum; and Δr is thewidth of the distribution; and

[0237] (b) The magnetic field is $\begin{matrix}{{B_{z} = {{- B_{c}} - {\frac{2\pi \quad I_{p}}{c}{\tanh \left( \frac{r^{2} - r_{0}^{2}}{r_{0}\Delta \quad r} \right)}}}},} & (75)\end{matrix}$

[0238] where B_(c) is the external field produced by the outer coil 325.Initially, B_(c)=B₀. This solution satisfies the boundary conditionsthat r=r_(a), and r=r_(b) are conductors (B_(normal)=0) andequipotentials with potential Φ=0. The boundary conditions are satisfiedif r₀ ²=(r_(a) ²+r_(b) ²)/2. r_(a)=10 cm and r₀=20 cm, so it followsthat r_(b)=26.5 cm. I_(p) is the plasma current per unit length.

[0239] The average velocities of the beam particles are V_(i)=r₀ω_(i)and V_(e)=r₀ω_(e), which are related by the equilibrium condition:$\begin{matrix}{{\omega_{e} = {\omega_{i}\left( {1 - \frac{\omega_{i}}{\Omega_{i}}} \right)}},} & (76)\end{matrix}$

[0240] where Ω_(i)=eB_(c)/(m_(i)c). Initially, it is assumed B_(c)=B₀,ω_(i)=Ω_(i)and ω_(e)=0. (In the initial equilibrium there is an electricfield such that the {right arrow over (E)}×{right arrow over (B)} andthe ΔB×{right arrow over (B)} drifts cancel. Other equilibrium arepossible according to the choice of B_(c).) The equilibrium equationsare assumed to be valid if ω_(i) and B_(c) are slowly varying functionsof time, but r₀=V_(i)/Ω_(i) remains constant.

[0241] The condition for this is the same as Eq. 66. Eq. 67 is alsosimilar, but the flux function Ψ has an additional term, i.e., Ψ=πr₀²(B_(z)) where $\begin{matrix}{{{\langle B_{z}\rangle} = {{\overset{\_}{B}}_{z} + {\frac{2\pi}{c}{I_{p}\left( \frac{r_{b}^{2} - r_{a}^{2}}{r_{b}^{2} + r_{a}^{2}} \right)}}}}{a\quad n\quad d}} & (77) \\{{\overset{\_}{B}}_{z} = {{- {B_{F}\left( \frac{r_{a}}{r_{0}} \right)}^{2}} - {{B_{c}\left\lbrack {1 - \left( \frac{r_{a}}{r_{0}} \right)^{2}} \right\rbrack}.}}} & (78)\end{matrix}$

[0242] The magnetic energy per unit length due to the beam current is$\begin{matrix}{{{\int_{r_{a}}^{r_{b}}{2\pi \quad r{{r\left( \frac{B_{z} - B_{c}}{8\pi} \right)}^{2}}}} = {\frac{1}{2}L_{p}I_{p}^{2}}},} & (79) \\{f\quad r\quad o\quad m\quad w\quad h\quad i\quad c\quad h} & \quad \\{L_{p} = {\frac{r_{b}^{2} - r_{a}^{2}}{r_{b}^{2} + r_{a}^{2}}\frac{2\pi^{2}r_{0}^{2}}{c^{2}}\quad a\quad n\quad d}} & \quad \\{{\langle B_{z}\rangle} = {{\overset{\_}{B}}_{z} + {\frac{c}{\pi \quad r_{0}^{2}}L_{p}{I_{p}.}}}} & (80)\end{matrix}$

[0243] The betatron condition of Eq. 70 is thus modified so that$\begin{matrix}{{\frac{\partial{\overset{\_}{B}}_{z}}{\partial t} = {{{- 2}\frac{\partial B_{c}}{\partial t}} - {\frac{L_{p}c}{\pi \quad r_{0}^{2}}\frac{\partial I_{p}}{\partial t}}}},} & (81) \\{{and}\quad {{Eq}.\quad 67}\quad {{becomes}:}} & \quad \\{\frac{\partial V_{i}}{\partial t} = {{\frac{e}{m_{i}}\frac{r_{0}}{c}\frac{\partial B_{c}}{\partial t}} = {{{- \frac{e}{2m_{i}c}}r_{0}\frac{\partial{\overset{\_}{B}}_{z}}{\partial t}} - {\frac{e}{m_{i}}\frac{L_{p}}{2\pi \quad r_{0}}{\frac{\partial I_{p}}{\partial t}.}}}}} & (82) \\{{{After}\quad {integrating}},} & \quad \\{{\Delta {\overset{\_}{B}}_{z}} = {{- 2}{{{B_{0}\left\lbrack {1 + \frac{r_{b}^{2} - r_{a}^{2}}{r_{0}^{2}}} \right\rbrack}\left\lbrack {\sqrt{\frac{W}{W_{0}}} - 1} \right\rbrack}.}}} & (83)\end{matrix}$

[0244] For W₀=100 eV and W=100 keV, Δ{right arrow over (B)}_(z)=−7.49kG. Integration of Eqs. 81 and 82 determines the value of the magneticfield produced by the field coil: $\begin{matrix}{B_{c} = {{B_{0}\sqrt{\frac{W}{W_{0}}}} = {2.19\quad {kG}}}} & (84) \\{and} & \quad \\{B_{F} = {{B_{F0} - {\left( \frac{r_{0}}{r_{a}} \right)^{2}\Delta \quad {\overset{\_}{B}}_{z}} - {\left( \frac{r_{0}^{2} - r_{a}^{2}}{r_{a}^{2}} \right)\Delta \quad B_{c}}} = {25\quad {{kG}.}}}} & (85)\end{matrix}$

[0245] If the final energy is 200 keV, B_(c)=3.13 kG and B_(F)=34.5 kG.The magnetic energy in the flux coil would be${\frac{B_{F}^{2}}{8\pi}\pi \quad r_{F}^{2}l} = {172\quad k\quad {J.}}$

[0246] The plasma current is initially 0.225 kA/cm corresponding to amagnetic field of 140 G, which increases to 10 kA/cm and a magneticfield of 6.26 kG. In the above calculations, the drag due to Coulombcollisions has been neglected. In the injection/trapping phase, it wasequivalent to 0.38 volts/cm. It decreases as the electron temperatureincreases during acceleration. The inductive drag, which is included, is4.7 volts/cm, assuming acceleration to 200 keV in 100 μs.

[0247] The betatron flux coil 320 also balances the drag from collisionsand inductance. The frictional and inductive drag can be described bythe equation: $\begin{matrix}{{\frac{\partial V_{b}}{\partial t} = {{- {V_{b}\left\lbrack {\frac{1}{t_{be}} + \frac{1}{t_{bi}}} \right\rbrack}} - {\frac{e}{m_{b}}\frac{L}{2\quad r_{0}}\frac{\partial I_{b}}{\partial t}}}},} & (86)\end{matrix}$

[0248] where (T_(i)/m_(i))<V_(b)<(T_(c)/m). Here, V_(b) is the beamvelocity, T_(e) and T_(i) are electron and ion temperatures, I_(b) isthe beam ion current, and$L = {{0.01257{r_{0}\left\lbrack {{\ln \left( \frac{8r_{0}}{a} \right)} - \frac{7}{4}} \right\rbrack}} = {0.71µ\quad H}}$

[0249] is the ring inductance. Also, r₀=20 cm and a=4 cm.

[0250] The Coulomb drag is determined by $\begin{matrix}{{t_{be} = {{\frac{3}{4}\sqrt{\frac{2}{\pi}}\left( \frac{m_{i}}{m} \right)\frac{T_{e}^{3/2}}{{ne}^{4}\ln \quad \Lambda}} = {195\quad µ\quad \sec}}}{t_{bi} = {\frac{2\sqrt{2m_{i}}W_{b}^{3/2}}{4\pi \quad {ne}^{4}\ln \quad \Lambda} = {54.8\quad µ\quad \sec}}}} & (87)\end{matrix}$

[0251] To compensate the drag, the betatron flux coil 320 must providean electric field of 1.9 volts/cm (0.38 volts/cm for the Coulomb dragand 1.56 volts/cm for the inductive drag). The magnetic field in thebetatron flux coil 320 must increase by 78 Gauss/μs to accomplish this,in which case V_(b) will be constant. The rise time of the current to4.5 kA is 18 μs, so that the magnetic field B_(F) will increase by 1.4kG. The magnetic field energy required in the betatron flux coil 320 is$\begin{matrix}{{\frac{B_{F}^{2}}{8\pi} \times \quad r_{F}^{2}l} = {394\quad {Joules}\quad {\left( {l = {115\quad {cm}}} \right).}}} & (88)\end{matrix}$

[0252] Betatron Formation Technique

[0253] Another preferred method of forming a FRC within the confinementsystem 300 is herein termed the betatron formation technique. Thistechnique is based on driving the betatron induced current directly toaccelerate a circulating plasma beam 335 using the betatron flux coil320. A preferred embodiment of this technique uses the confinementsystem 300 depicted in FIG. 25, except that the injection of low energyion beams is not necessary.

[0254] As indicated, the main component in the betatron formationtechnique is the betatron flux coil 320 mounted in the center and alongthe axis of the chamber 310. Due to its separate parallel windingsconstruction, the coil 320 exhibits very low inductance and, whencoupled to an adequate power source, has a low LC time constant, whichenables rapid ramp up of the current in the flux coil 320.

[0255] Preferably, formation of the FRC commences by energizing theexternal field coils 325, 330. This provides an axial guide field aswell as radial magnetic field components near the ends to axiallyconfine the plasma injected into the chamber 310. Once sufficientmagnetic field is established, the background plasma sources 345 areenergized from their own power supplies. Plasma emanating from the gunsstreams along the axial guide field and spreads slightly due to itstemperature. As the plasma reaches the mid-plane of the chamber 310, acontinuous, axially extending, annular layer of cold, slowly movingplasma is established.

[0256] At this point the betatron flux coil 320 is energized. Therapidly rising current in the coil 320 causes a fast changing axial fluxin the coil's interior. By virtue of inductive effects this rapidincrease in axial flux causes the generation of an azimuthal electricfield E (see FIG. 29), which permeates the space around the flux coil.By Maxwell's equations, this electric field is directly proportional tothe change in strength of the magnetic flux inside the coil, i.e.: afaster betatron coil current ramp-up will lead to a stronger electricfield.

[0257] The inductively created electric field couples to the chargedparticles in the plasma and causes a ponderomotive force, whichaccelerates the particles in the annular plasma layer. Electrons, byvirtue of their smaller mass, are the first species to experienceacceleration. The initial current formed by this process is, thus,primarily due to electrons. However, sufficient acceleration time(around hundreds of micro-seconds) will eventually also lead to ioncurrent. Referring to FIG. 29, this electric field accelerates theelectrons and ions in opposite directions. Once both species reach theirterminal velocities, current is carried about equally by ions andelectrons.

[0258] As noted above, the current carried by the rotating plasma givesrise to a self magnetic field. The creation of the actual FRC topologysets in when the self magnetic field created by the current in theplasma layer becomes comparable to the applied magnetic field from theexternal field coils 325, 330. At this point magnetic reconnectionoccurs and the open field lines of the initial externally producedmagnetic field begin to close and form the FRC flux surfaces (see FIGS.8 and 10).

[0259] The base FRC established by this method exhibits modest magneticfield and particle energies that are typically not at reactor relevantoperating parameters. However, the inductive electric acceleration fieldwill persist, as long as the current in the betatron flux coil 320continues to increase at a rapid rate. The effect of this process isthat the energy and total magnetic field strength of the FRC continuesto grow. The extent of this process is, thus, primarily limited by theflux coil power supply, as continued delivery of current requires amassive energy storage bank. However, it is, in principal,straightforward to accelerate the system to reactor relevant conditions.

[0260] For field reversal, the circulating plasma beam 335 is preferablyaccelerated to a rotational energy of about 100 eV, and preferably in arange of about 75 eV to 125 eV. To reach fusion relevant conditions, thecirculating plasma beam 335 is preferably accelerated to about 200 keVand preferably to a range of about 100 keV to 3.3 MeV. When ion beamsare added to the circulating plasma beam 335, as described above, theplasma beam 335 depolarizes the ion beams.

[0261] Experiments—Beam Trapping and FRC Formation

[0262] Experiment 1: Propagating and trapping of a neutralized beam in amagnetic containment vessel to create an FRC.

[0263] Beam propagation and trapping were successfully demonstrated atthe following parameter levels:

[0264] Vacuum chamber dimensions: about 1 m diameter, 1.5 m length.

[0265] Betatron coil radius of 10 cm.

[0266] Plasma beam orbit radius of 20 cm.

[0267] Mean kinetic energy of streaming beam plasma was measured to beabout 100 eV, with a density of about 103 cm⁻³, kinetic temperature onthe order of 10 eV and a pulse-length of about 20 μs.

[0268] Mean magnetic field produced in the trapping volume was around100 Gauss, with a ramp-up period of 150 μs. Source: Outer coils andbetatron coils.

[0269] Neutralizing background plasma (substantially Hydrogen gas) wascharacterized by a mean density of about 10¹³ cm⁻³, kinetic temperatureof less than 10 eV.

[0270] The beam was generated in a deflagration type plasma gun. Theplasma beam source was neutral Hydrogen gas, which was injected throughthe back of the gun through a special puff valve. Different geometricaldesigns of the electrode assembly were utilized in an overallcylindrical arrangement. The charging voltage was typically adjustedbetween 5 and 7.5 kV. Peak breakdown currents in the guns exceeded250,000 A. During part of the experimental runs, additional pre-ionizedplasma was provided by means of an array of small peripheral cable gunsfeeding into the central gun electrode assembly before, during or afterneutral gas injection. This provided for extended pulse lengths of above25 μs.

[0271] The emerging low energy neutralized beam was cooled by means ofstreaming through a drift tube of non-conducting material beforeentering the main vacuum chamber. The beam plasma was alsopre-magnetized while streaming through this tube by means of permanentmagnets.

[0272] The beam self-polarized while traveling through the drift tubeand entering the chamber, causing the generation of a beam-internalelectric field that offset the magnetic field forces on the beam. Byvirtue of this mechanism it was possible to propagate beams ascharacterized above through a region of magnetic field withoutdeflection.

[0273] Upon further penetration into the chamber, the beam reached thedesired orbit location and encountered a layer of background plasmaprovided by an array of cable guns and other surface flashover sources.The proximity of sufficient electron density caused the beam to looseits self-polarization field and follow single particle like orbits,essentially trapping the beam. Faraday cup and B-dot probe measurementsconfirmed the trapping of the beam and its orbit. The beam was observedto have performed the desired circular orbit upon trapping. The beamplasma was followed along its orbit for close to ¾ of a turn. Themeasurements indicated that continued frictional and inductive lossescaused the beam particles to loose sufficient energy for them to curlinward from the desired orbit and hit the betatron coil surface ataround the {fraction (3/4 )} turn mark. To prevent this, the lossescould be compensated by supplying additional energy to the orbiting beamby inductively driving the particles by means of the betatron coil.

[0274] Experiment 2: FRC formation utilizing the combined beam/betatronformation technique.

[0275] FRC formation was successfully demonstrated utilizing thecombined beam/betatron formation technique. The combined beam/betatronformation technique was performed experimentally in a chamber 1 m indiameter and 1.5 m in length using an externally applied magnetic fieldof up to 500 G, a magnetic field from the betatron flux coil 320 of upto 5 kG, and a vacuum of 1.2×10⁻⁵ torr. In the experiment, thebackground plasma had a density of 10¹³ cm⁻³ and the ion beam was aneutralized Hydrogen beam having a density of 1.2×10¹³ cm⁻³, a velocityof 2×10⁷ cm/s, and a pulse length of around 20 μs (at half height).Field reversal was observed.

[0276] Experiment 3: FRC formation utilizing the betatron formationtechnique.

[0277] FRC formation utilizing the betatron formation technique wassuccessfully demonstrated at the following parameter levels:

[0278] Vacuum chamber dimensions: about 1 m diameter, 1.5 m length.

[0279] Betatron coil radius of 10 cm.

[0280] Plasma-orbit radius of 20 cm.

[0281] Mean external magnetic field produced in the vacuum chamber wasup to 100 Gauss, with a ramp-up period of 150 μs and a mirror ratio of 2to 1. (Source: Outer coils and betatron coils).

[0282] The background plasma (substantially Hydrogen gas) wascharacterized by a mean density of about 10¹³ cm⁻³, kinetic temperatureof less than 10 eV.

[0283] The lifetime of the configuration was limited by the total energystored in the experiment and generally was around 30 μs.

[0284] The experiments proceeded by first injecting a background plasmalayer by two sets of coaxial cable guns mounted in a circular fashioninside the chamber. Each collection of 8 guns was mounted on one of thetwo mirror coil assemblies. The guns were azimuthally spaced in anequidistant fashion and offset relative to the other set. Thisarrangement allowed for the guns to be fired simultaneously and therebycreated an annular plasma layer.

[0285] Upon establishment of this layer, the betatron flux coil wasenergized. Rising current in the betatron coil windings caused anincrease in flux inside the coil, which gave rise to an azimuthalelectric field curling around the betatron coil. Quick ramp-up and highcurrent in the betatron flux coil produced a strong electric field,which accelerated the annular plasma layer and thereby induced asizeable current. Sufficiently strong plasma current produced a magneticself-field that altered the externally supplied field and caused thecreation of the field reversed configuration. Detailed measurements withB-dot loops identified the extent, strength and duration of the FRC.

[0286] An example of typical data is shown by the traces of B-dot probesignals in FIG. 30. The data curve A represents the absolute strength ofthe axial component of the magnetic field at the axial mid-plane (75 cmfrom either end plate) of the experimental chamber and at a radialposition of 15 cm. The data curve B represents the absolute strength ofthe axial component of the magnetic field at the chamber axial mid-planeand at a radial position of 30 cm. The curve A data set, therefore,indicates magnetic field strength inside of the fuel plasma layer(between betatron coil and plasma) while the curve B data set depictsthe magnetic field strength outside of the fuel plasma layer. The dataclearly indicates that the inner magnetic field reverses orientation (isnegative) between about 23 and 47 μs, while the outer field stayspositive, i.e., does not reverse orientation. The time of reversal islimited by the ramp-up of current in the betatron coil. Once peakcurrent is reached in the betatron coil, the induced current in the fuelplasma layer starts to decrease and the FRC rapidly decays. Up to nowthe lifetime of the FRC is limited by the energy that can be stored inthe experiment. As with the injection and trapping experiments, thesystem can be upgraded to provide longer FRC lifetime and accelerationto reactor relevant parameters.

[0287] Overall, this technique not only produces a compact FRC, but itis also robust and straightforward to implement. Most importantly, thebase FRC created by this method can be easily accelerated to any desiredlevel of rotational energy and magnetic field strength. This is crucialfor fusion applications—and classical confinement of high-energy fuelbeams.

[0288] Experiment 4: FRC formation utilizing the betatron formationtechnique.

[0289] An attempt to form an FRC utilizing the betatron formationtechnique has been performed experimentally in a chamber 1 m in diameterand 1.5 m in length using an externally applied magnetic field of up to500 G, a magnetic field from the betatron flux coil 320 of up to 5 kG,and a vacuum of 5×10⁻⁶ torr. In the experiment, the background plasmacomprised substantially Hydrogen with of a density of 10¹³ cm⁻³ and alifetime of about 40 μs. Field reversal was observed.

[0290] Fusion

[0291] Significantly, these two techniques for forming a FRC inside of acontainment system 300 described above, or the like, can result inplasmas having properties suitable for causing nuclear fusion therein.More particularly, the FRC formed by these methods can be accelerated toany desired level of rotational energy and magnetic field strength. Thisis crucial for fusion applications and classical confinement ofhigh-energy fuel beams. In the confinement system 300, therefore, itbecomes possible to trap and confine high-energy plasma beams forsufficient periods of time to cause a fusion reaction therewith.

[0292] To accommodate fusion, the FRC formed by these methods ispreferably accelerated to appropriate levels of rotational energy andmagnetic field strength by betatron acceleration. Fusion, however, tendsto require a particular set of physical conditions for any reaction totake place. In addition, to achieve efficient bum-up of the fuel andobtain a positive energy balance, the fuel has to be kept in this statesubstantially unchanged for prolonged periods of time. This isimportant, as high kinetic temperature and/or energy characterize afusion relevant state. Creation of this state, therefore, requiressizeable input of energy, which can only be recovered if most of thefuel undergoes fusion. As a consequence, the confinement time of thefuel has to be longer than its bum time. This leads to a positive energybalance and consequently net energy output.

[0293] A significant advantage of the present invention is that theconfinement system and plasma described herein are capable of longconfinement times, i.e., confinement times that exceed fuel bum times. Atypical state for fusion is, thus, characterized by the followingphysical conditions (which tend to vary based on fuel and operatingmode):

[0294] Average ion temperature: in a range of about 30 to 230 keV andpreferably in a range of about 80 keV to 230 keV

[0295] Average electron temperature: in a range of about 30 to 100 keVand preferably in a range of about 80 to 100 keV

[0296] Coherent energy of the fuel beams (injected ion beams andcirculating plasma beam): in a range of about 100 keV to 3.3 MeV andpreferably in a range of about 300 keV to 3.3 MeV.

[0297] Total magnetic field: in a range of about 47.5 to 120 kG andpreferably in a range of about 95 to 120 kG (with the externally appliedfield in a range of about 2.5 to 15 kG and preferably in a range ofabout 5 to 15 kG).

[0298] Classical Confinement time: greater than the fuel burn time andpreferably in a range of about 10 to 100 seconds.

[0299] Fuel ion density: in a range of about 10¹⁴ to less than 10¹⁶ cm⁻³and preferably in a range of about 10¹⁴ to 10¹⁵ cm³.

[0300] Total Fusion Power: preferably in a range of about 50 to 450kW/cm (power per cm of chamber length)

[0301] To accommodate the fusion state illustrated above, the FRC ispreferably accelerated to a level of coherent rotational energypreferably in a range of about 100 keV to 3.3 MeV, and more preferablyin a range of about 300 keV to 3.3 MeV, and a level of magnetic fieldstrength preferably in a range of about 45 to 120 kG, and morepreferably in a range of about 90 to 115 kG. At these levels, highenergy ion beams can be injected into the FRC and trapped to form aplasma beam layer wherein the plasma beam ions are magnetically confinedand the plasma beam electrons are electrostatically confined.

[0302] Preferably, the electron temperature is kept as low aspractically possible to reduce the amount of bremsstrahlung radiation,which can, otherwise, lead to radiative energy losses. The electrostaticenergy well of the present invention provides an effective means ofaccomplishing this.

[0303] The ion temperature is preferably kept at a level that providesfor efficient burn-up since the fusion cross-section is a function ofion temperature. High direct energy of the fuel ion beams is essentialto provide classical transport as discussed in this application. It alsominimizes the effects of instabilities on the fuel plasma. The magneticfield is consistent with the beam rotation energy. It is partiallycreated by the plasma beam (self-field) and in turn provides the supportand force to keep the plasma beam on the desired orbit.

[0304] Fusion Products

[0305] The fusion products are born predominantly near the null surfacefrom where they emerge by diffusion towards the separatrix 84 (see FIG.8). This is due to collisions with electrons (as collisions with ions donot change the center of mass and therefore do not cause them to changefield lines). Because of their high kinetic energy (product ions havemuch higher energy than the fuel ions), the fusion products can readilycross the separatrix 84. Once they are beyond the separatrix 84, theycan leave along the open field lines 80 provided that they experiencescattering from ion-ion collisions. Although this collisional processdoes not lead to diffusion, it can change the direction of the ionvelocity vector such that it points parallel to the magnetic field.These open field lines 80 connect the FRC topology of the core with theuniform applied field provided outside the FRC topology. Product ionsemerge on different field lines, which they follow with a distributionof energies; advantageously in the form of a rotating annular beam. Inthe strong magnetic fields found outside the separatrix 84 (typicallyaround 100 kG), the product ions have an associated distribution ofgyro-radii that varies from a minimum value of about 1 cm to a maximumof around 3 cm for the most energetic product ions.

[0306] Initially the product ions have longitudinal as well asrotational energy characterized by ½ M(v_(par))² and ½ M(v_(perp))².v_(perp) is the azimuthal velocity associated with rotation around afield line as the orbital center. Since the field lines spread out afterleaving the vicinity of the FRC topology, the rotational energy tends todecrease while the total energy remains constant. This is a consequenceof the adiabatic invariance of the magnetic moment of the product ions.It is well known in the art that charged particles orbiting in amagnetic field have a magnetic moment associated with their motion. Inthe case of particles moving along a slow changing magnetic field, therealso exists an adiabatic invariant of the motion described by ½M(v_(perp))²/B. The product ions orbiting around their respective fieldlines have a magnetic moment and such an adiabatic invariant associatedwith their motion. Since B decreases by a factor of about 10 (indicatedby the spreading of the field lines), it follows that v_(perp) willlikewise decrease by about 3.2. Thus, by the time the product ionsarrive at the uniform field region their rotational energy would be lessthan 5% of their total energy; in other words almost all the energy isin the longitudinal component.

[0307] While the invention is susceptible to various modifications andalternative forms, a specific example thereof has been shown in thedrawings and is herein described in detail. It should be understood,however, that the invention is not to be limited to the particular formdisclosed, but to the contrary, the invention is to cover allmodifications, equivalents, and alternatives falling within the spiritand scope of the appended claims.

What is claimed is:
 1. A method of forming a magnetic field of fieldreverse topology comprising the steps of injecting a plasma into achamber, applying a magnetic field to form a first magnetic field in thechamber having unidirectional field lines, injecting ion beams into thechamber substantially transverse to the first magnetic field, trappingthe ion beams in betatron orbits within the first magnetic field,forming a rotating plasma beam within the chamber having a current andan internal magnetic field due to the first magnetic field, forming apoloidal second magnetic field about the rotating plasma having externalfield lines outside the rotating plasma extending in a same direction asthe field lines of the first magnetic field and internal field linesextending in an opposite direction to the field lines of the firstmagnetic field, injecting a current through a betatron flux coil in thechamber, inducing an azimuthal electric field inside the chamber,increasing the rotating plasma beam's rotational velocity, increasingthe second magnetic field's magnitude beyond the magnitude of the firstmagnetic field, and reversing the direction of the internal field withinthe rotating plasma and forming a combined magnetic field of fieldreverse topology (FRC).
 2. The method of claim 1 wherein the ion beamsare injected substantially transverse to the first magnetic field. 3.The method of claim 2 further comprising the step of neutrilizing theion beams.
 4. The method of claim 3 further comprising the step ofexerting a Lorentz force due to the first magnetic field on theneutralized ion beams to bend the ion beams into betatron orbits.
 5. Themethod of claim 4 further comprising the step of draining theneutralized ion beams' electric polarization.
 6. The method of claim 1further comprising the step of maintaining the rotating beam plasma at apredetermined radial size.
 7. The method of claim 6 further comprisingthe step of increasing the first magnetic field's magnitude.
 8. Themethod of claim 1 further comprising the step of accelerating therotating plasma beam to a fusion relevant rotational energy.
 9. Themethod of claim 8 further comprising the steps of injecting high energyion beams into the FRC and trapping the beams in betatron orbits withinthe FRC.
 10. The method of forming a field reversed configurationmagnetic field within a reactor chamber comprising the steps of applyinga magnetic field to a reactor chamber in which plasma is filled,injecting ion beams into the applied magnetic field within the reactorchamber, forming a rotating plasma beam within the chamber having apoloidal magnetic self-field, and increasing the rotating plasma beam'srotational velocity to increase the magnetic self-field's magnitudebeyond the applied magnetic field's magnitude causing field reversalinternal to the rotating plasma beam and formation of a combinedmagnetic field having a field reverse configuration (FRC).
 11. Themethod of claim 10 wherein the step of applying a magnetic fieldincludes energizing a plurality of field coils extending about thechamber.
 12. The method of claim wherein the ion beams are injectedsubstantially transverse to the applied magnetic field.
 13. The methodof claim 12 wherein the step of injecting the ion beams furthercomprises the steps of neutrilizing the ion beams, draining theneutralized ion beams' electric polarization, and exerting a Lorentzforce due to the first magnetic field on the neutralized ion beams tobend the ion beams into betatron orbits.
 14. The method of claim 10further comprising the step of increasing the applied magnetic field'smagnitude to maintain the rotating beam plasma at a predetermined radialsize.
 15. The method of claim 10 wherein step of increasing the rotatingplasma beam's rotational velocity includes the step of energizing abetatron flux coil within the chamber inducing an azimuthal electricfield within the chamber.
 16. The method of claim 15 further comprisingthe step increasing the current through the flux coil to accelerate therotating plasma beam to a fusion relevant rotational energy.
 17. Themethod of claim 16 further comprising the steps of injecting high energyion beams into the FRC and trapping the beams in betatron orbits withinthe FRC.